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/ 

AN 



INTRODUCTION 



fO^T- 



ASTRONOMY: 



DESIGNED AS A 



TEXT-BOOK 



FOE THE USE OF 



STUDENTS IK COLLEGE, 



U BY 
DENISON OLMSTED, LLD., 

LATE PROFESSOR OF NATURAL PHILOSOPHY AND ASTRONOMY IN 
YALE COLLEGE. 



REVISED 

By E. S. SNELL, LL.D., 

PROFESSOR OF MATHEMATICS AND NATURAL PHILOSOPHY IN 
AMHERST COLLEGE. 



NEW YORK: 
COLLINS & BROTHER, 

NO. 84 LEONARD STREET. 
1863. 









7492 



Entered according to Act of Congress, in the year 1844, 

By DENISON OLMSTED, 

In the Clerk's Office of the District Court of Connecticut. 



Revised Edition. 

Entered according to Act of Congress, in the year 1861, 

By JULIA M. OLMSTED, 

Foe the Children op Denison Olmsted, deceased, 

In the Clerk's Office of the District Court of the District of Connecticut. 



Rennie, Shea & Lindsay, 
stereotypers and electrotypers, 

81, 83, & 85 CENTRE-STREET, 
Nefio goi-ft. 

C. A. AI/VORD, 

Printer, 

15 Vandewater-street, New York. 



PREFACE TO THE REVISED EDITION. 



This work was revised by its author in 1S54, at which time 
were introduced notices of the recent discoveries in Astronomy. 
The changes made in the present edition are mostly of a differ- 
ent character. "While there have been added brief accounts of 
still later discoveries, and of new methods of observation, it 
has been especially my aim to give more clearness to certain 
descriptions and demonstrations, which my experience as a 
teacher has shown me to be perplexing to the learner. The 
discussion of central forces is entirely remodeled, and the great 
law of gravitation throughout the solar system is deduced, by 
a more direct and less cumbrous mode of reasoning, from the 
three laws of Kepler, which are taken as facts established by 
observation. 

To those numerous teachers who approve the general design 
of Professor Olmsted in the preparation of this work, as set 
forth in his preface, and who have themselves tested the utility 
of that design, it is hoped that the changes now made will 
commend themselves as improvements. 

E. S. SNELL. 
Amherst College, March, 1861. 



AUTHOR'S PREFACE. 



Nearly all who have written Treatises on Astronomy, designed for 
young learners, appear to have erred in one of two ways ; they have 
either disregarded demonstrative evidence, and relied on mere popular 
illustration, or they have exhibited the elements of the science in naked 
mathematical formulas. The former are usually diffuse and superficial ; 
the latter, technical and abstruse. 

In the following Treatise, we have endeavored to unite the advantages 
of both methods. We have sought, first, to establish the great princi- 
ples of astronomy on a mathematical basis ; and, secondly, to render the 
study interesting and intelligible to the learner, by easy and familiar 
illustrations. We would not encourage any one to believe that he can 
enjoy a full view of the grand edifice of astronomy, while its noble 
foundations are hidden from his sight ; nor would we assure him that 
he can contemplate the structure in its true magnificence, while its 
basement alone is within his field of vision. W T e would, therefore, that 
the student of astronomy should confine his attention neither to the ex- 
terior of the building, nor to the mere analytic investigation of its struc- 
ture. We would desire that he should not only study it in models and 
diagrams, and mathematical formulas, but should at the same time 
acquire a love of nature herself, and cultivate the habit of raising his 
views to the grand originals. Nor is the effort to form a clear concep- 
tion of the motions and dimensions of the heavenly bodies, less favorable 
to the improvement of the intellectual powers, than the study of pure 
geometry. 

But it is evidently possible to follow oat all the intricacies of an ana- 
lytical process, and to arrive at a full conviction of the great truths of 
astronomy, and yet know very little of nature. According to our expe- 
rience, however, but few students in the course of a liberal education 
will feel satisfied with this. They do not need so much to be convinced 
that the assertions of astronomers are true, as they desire to know what 
the truths are, and how they were ascertained ; and they will derive 



VI 

from the stud)' of astronomy little of that moral and intellectual eleva- 
tion which they had anticipated, unless they learn to look upon the 
heavens with new views, and a clear comprehension of their wonderful 
mechanism. 

Much of the difficulty that usually attends the early progress of the 
astronomical student, arises from his being too soon introduced to the 
most perplexing part of the whole subject,— the planetary motions. In 
this work, the consideration of these is for the most part postponed until 
the learner has become familiar with the artificial circles of the sphere, 
and conversant with the celestial bodies. We then first take the most 
simple view possible of the planetary motions by contemplating them as 
they really are in nature, and afterwards proceed to the more difficult 
inquiry, why they appear as they do. Probably no science derives such 
signal advantage from a happy arrangement, as astronomy ; — an order, 
which brings out every fact or doctrine of the science just in the place 
where the mind of the learner is prepared to receive it. 



ANALYSIS. 



PRELIMINARY OBSERVATIONS. 

Article. 

Astronomy, and its divisions 1 

Descriptive, physical, practical 1 

History. — Ancient nations who culti- 
vated astronomy 2 

Pythagoras — his time — his views. ... 2 

Alexandrian school — Hipparchus. . . . 2 

Ptolemy — the Almagest 2 

Copernicus, Tycho Brahe, Kepler, Ga- 
lileo, Newton, La Place 2 

Astrology — Natural— Judicial 8 

Accuracy aimed at in astronomy. .... 4 
Truths not always to be proved when 

stated 5 

Coperniean system — its doctrines. ... 6 

Plan of the work 7 



PAET I. THE EARTH. 
Chapter I. — Of the figure and size of 

THE EARTH, AND THE DOCTRINE OF THE 
SPHERE. 

Figure of the earth — proofs 8 

Dip of the horizon 9 

Its relation to height — table 10 

The exact form 11 

Dimensions of the earth 12 

How found 13, 14 

Erroneous ideas of up and down 15 

Doctrine of the sphere 16 

Sections by a plane 17 

Axis of a circle — poles 18, 19 

Great circles bisect each other 20 

Secondaries 21 

Measure of inclination. 22 

Terrestrial and celestial spheres ..... 23 

Horizon— rational and sensible 24 

Zenith and Nadir. 25 

Vertical circles — meridian, prime ver- 
tical , 26 

Co-ordinates for the horizon, ampli- 
tude, or azimuth — altitude, or ze- 
nith distance 27 

Axis and poles of the earth 28 

Equator— equinoctial 29 

Its secondaries— the meridians, or 

hour circles 30 

Latitude — polar distance 31 

Longitude , , 32 

The ecliptic — inclination to the equa- 
tor 33 

Vernal and autumnal equinoxes 34 

Solstices— signs of the zodiac 85 



Article. 

Colures — equinoctial, solstitial 36 

Co-ordinates to the equinoctial — right 

ascension, declination 37 

Co-ordinates to the ecliptic, celestial 

longitude and latitude 37 

Parallels of latitude 38 

Tropics 39 

Polar circles 40 

Zones 41 

Zodiac 42 

Elevation of the pole 43 

Elevation of the equator 44 

Polar distance 45 

Chapter II. — Diurnal revolution — Arti- 
ficial GLOBES — ASTRONOMICAL PROBLEMS, 

Circles of Diurnal Ee volution 46 

Sidereal day defined 47 

Appearance of the circles of diurnal 

revolution at the equator 49 

A Right Sphere defined 49 

A Parallel Sphere 50-52 

An Oblique Sphere 53 

Circle of Perpetual Apparition ...... 54 

Circle of Perpetual Occupation 55 

How are the circles of daily motion 
cut by the horizon in the different 

spheres 56 

Explanation of the peculiar appear- 
ances of each sphere, from the revo- 
lution of the earth on its axis. .. .57-60 
Artificial Globes — terrestrial and ce- 
lestial 61 

Meridian — how represented — how 

graduated 62 

Horizon — how represented — how gra- 
duated 62 

Hour Circles — how represented 63 

Hour Index described 64 

Quadrant of Altitude — its use 65 

To rectify the globe for any place. ... 66 
Problems on thk Terrestrial Globe 
—To find the latitude and longitude 

of a place 67 

To find & place, its latitude and longi- 
tude being given 68 

To find the bearing and distance of 

two places 69 

To determine the difference of time of 

two places 70 

The hour being given at any place, to 
tell what hour it is in any "other part 

of the world 71 

To find the antoeci, periceci, and antipo- 
des 72 



Vlll 



ANALYSIS. 



Art. 
To rectify the globe for the sun's 
place 73 

The latitude of the place being given, 
to find the time of the sun's rising 
and setting 74 

Problem.? on the Celestial Globe. — 

To find the right ascension and decli- 
nation ... 75 

To represent the appearance of the 
heavens at any time 76 

To find the altitude and azimuth of a 
star , 77 

To find the angular distance of two 
stars from each other 78 

To find the sun's meridian altitude, 
the latitude and day of the month 
being given 79 

Chapter III. — Parallax — Refraction — 
Twilight. 

Parallax defined — diurnal 80 

True place 81 

Relation of parallax to the zenith dis- 
tance, and distance from the center 

of the earth 82 

To find the horizontal parallax from 

the parallax at any altitude 83 

Amount of parallax in the zenith and 

in the horizon 83 

Effect of parallax upon the altitude of 

a body 84 

Mode of determining the horizontal 

parallax of a body 85 

Amount of the sun's hor. par 86 

Use of parallax 87 

Refraction. — Its effect upon the alti- 
tude of a body 88 

Its nature illustrated 88 

Its amount at different angles of ele- 
vation 89 

How the amount is ascertained 90-91 

Sources of inaccuracy in estimating 

the refraction 92 

Effect of refraction upon the sun and 

moon when near the horizon 93 

Oval figure of these bodies explained. 94 
Apparent enlargement of the sun and 

moon near the horizon 95 

Twilight.— its, cause explained 96 

Length of twilight in different lati- 
tudes 97 

How the atmosphere contributes to 
diffuse the sun's light 98 

Chapter IV.— Time. 

Time defined 99 

What period is a sidereal day 100 

Uniformity of sidereal days 100 

Solar time, how reckoned 101 

Why solar days are longer than side- 
real 101 

Apparent time defined 102 

Mean time 103 

An astronomical day 103 

Equation of time defined 104 

When do apparent time and mean 

time differ most 104 

When do they come together 104 



Art. 

Effect of a change in the place of the 
earth's perihelion 104 

Causes of the inequality of the solar 
days 105 

Explain the first cause, depending on 
the unequal velocities of the sun. . 105 

Explain the second cause, depending 
on the obliquity of the ecliptic. .106-108 

When does the sidereal day com- 
mence , 109 

The Calendar.— Astronomical year de- 
fined 110 

How the most ancient nations deter- 
mined the number of days in the 
year Ill 

Julius Caesar's reformation of the cal- 
endar explained Ill 

Errors of this calendar 112 

Reformation by Pope Gregory 112 

Rule for the Gregorian calendar 112 

New style, whenadopted in England. 113 

What nations still adhere to the old 
style 113 

What number of days is now allowed 
between old and new style 114 

How the common year begins and 
ends 115 

How leap year begins and ends 115 

Does the confusion of different calen- 
dars affect astronomical observations 116 

Chapter V. — Astronomical Instruments 
and Problems — Figure and Density 
of the Earth. 

How the most ancient nations acquired 

their knowledge of astronomy 117 

Use of Instruments in the Alexan- 
drian School 117 

Ditto, by Tycho Brahe 117 

Ditto, by the Astronomers Royal 117 

Space occupied by \" on the limb of 

an instrument 117 

Extent of actual divisions on the limb/ 118 

Vernier defined 118 

Its use illustrated ^ 119 

Chief astronomical instruments eau- 

merated 120 

Observations taken on the meridian. 120 

Reasons of this. 120 

Transit Instrument defined 121 

Ditto, described 121 

Method of placing it in the meridian. 122 

Line of collimation defined 123 

System of wires in the focus 123 

Astronomical Clock— how regulated . . 124 

What does it show 124 

How to test its accuracy 124 

Rate and error 124 

American method 125 

Mural Circle — its object 126 

Describe it 126 

How the different parts contribute to 

the object 126 

Use of the Mural Circle for arcs of 

declination 127 

Altitude and Azimuth Instrument de- 
fined 128 

Its use 12S 



ANALYSIS. 



IX 



Art. 

Describe it 128 

Sextant described 129 

How to measure the angular distance 

of the moon from a star. .■ 129 

How to take the altitude of a heavenly- 
body 129 

Use of the artificial horizon 130 

In what consists the peculiar value of 

the sextant . 130 

Astronomical Problems. — Given the 
sun's right ascension and declina- 
tion, to find his longitude and the 

obliquity of the ecliptic 132 

Napier's rule of circular parts 132 

Given the sun's declination to find his 
rising and setting at any place whose 

latitude is known 133 

Given the latitude of a place and the 
declination of a heavenly body, to 
determine its altitude and azimuth 
when on the six o'clock hour circle. 134 
The latitudes and longitudes of two 
celestial objects being given, to find 

their distance apart 135 

Figure and Density of the Earth — 
reason for ascertaining it with great 

precision 136 

How found from the centrifugal force. 137 
From measuring an arc of the meridian 138 
From observations with the pendulum 139 

From the motions of the moon 140 

From precession 141 

Density of the earth 141 

How ascertained by Dr. Maskelyne.. 141 
Why an important element 141 



Part II. -OF THE SOLAR SYSTEM. 

Chapter I. — The Sun — Solar Spots — Zo- 
diacal Light. 

Figure of the sun 143 

Angle subtended by a line of 400 miles 143 

Distance from the earth 144 

Illustrated by motion on a railway car. 144 
Apparent diameter of the sun — how 

found 145 

How to find the linear diameter 145 

How much larger is the sun than the 

earth 145 

Its density and mass compared with 

the earth's 146 

Weight at the surface of the sun 146 

Velocity of falling bodies at the sun. 146 

Solar Spots. — Their number 147 

Size 147 

Description 147 

What region of the sun do they occupy 147 

Proof that they are on the sun 148 

How we learn the revolution of the 

sun on his axis 148 

Time of the revolution 148 

Apparent paths of the spots 149 

Inclination of the solar axis 149 

Sun's Nodes — when does the sun pass 

them? 150 

Faculae 151 



Art. 

Theory of the spots 151 

Zodiacal Light. — Where seen 152 

Its form 152 

Aspects at different seasons 152 

1 ts motions 152 

Its nature 152 

Chapter II. — Apparent Annual Motion 
of the Sun — Seasons — Figure of thb 
Earth's Orbit. 

Apparent motion of the sun 153 

How both the sun and earth are said 

to move from west to east 154 

Nature and position of the sun's orbit, 

how determined 155 

Changes in declination, how found. . 155 

Ditto, in right ascension 156 

Inferences from a table of the sun's 

declinations 156 

Ditto, of right ascensions 156 

Path of the sun, how proved to be a 

great circle 156 

Obliquity of the ecliptic, how found. . 157 

How it varies 157 

Great dimensions of the earth's orbit. 158 

Earth's daily motion in miles 158 

Ditto, hourly, ditto 158 

Diurnal motion at the equator per hour 158 
Seasons. — Causes of the change of 

seasons 159 

How each cause operates 159 

Illustrated by a diagram 160 

Change of seasons had the equator 

been perpendicular to the ecliptic. . 161 
Figure of the Earth's Orbit. — Proof 
that the earth's orbit is not circular. 161 

Radius vector defined 162 

Figure of the earth's orbit, how ob- 
tained 162 

Relative distances of the earth from 

the sun, how found 163 

Perihelion and Aphelion defined 164 

Variations in the sun's apparent diam- 
eter 164 

Angular velocities of the sun at the 

perihelion and aphelion 165 

Ratio of these velocities to the dis- 
tances 165 

How to calculate the relative distances 
of the earth from the sun's daily mo- 
tions 165 

Product of the angle described in any 
given time by the square of the dis- 
tance 167 

Space described by the radius vector 

of the solar orbit in equal times. . . 168 
How to represent the sun's orbit by a 
diagram 168 

Chapter III. — Central Forces — Gravita- 
tion. 

Two forces in curved motion 170 

Centripetal and centrifugal 170 

Kepler's laws 171 

First law — second — third 171 

First principles in mechanics and as- 
tronomy identical 171 

Kepler's first law proved 172 



ANALYSTS. 



Art. 
Its converse 173 

Measure of velocity in an orbit 174 

Modification for those nearly circular. 174 

Centrifugal force 175 

First law of central force in circular 

orl.it 176 

Second ditto 177 

Law of gravity in an elliptical orbit. . 178 

Proof. 179 

Same law for different orbits— proof. . 181 
Bodies on the earth compared with 

the moon 182 

Same law explains disturbances 182 

Gravity as quantity of matter. . .... 182 

Projectiles 183 

"Why parabola, not ellipse 183 

Time of complete revolution of a pro- 
jectile 183 

Different forms of projectile paths. . . 184 

Place of perigee and apogee 184 

Annual and diurnal rotation by one 

impulse 185 

Effect of impulse on the system 186 

Two bodies supposed 186 

Motion of center 186 

Motion of each body 186 

Epicycloids, two forms 186 

Where retrograde motion 186 

How center can be kept at rest 186 

Why planet returns from aphelion. . . 187 

Why it departs from perihelion 187 

Change in centrifugal force, compared 
with that in centripetal 187 

Chapter IV. — Precession of the Equi- 
noxes — Nutation — Aberration — Mean 
and True Places of the Sun. 

Precession of the equinoxes defined. . 188 

Why so called 188 

Amount of precession annually 189 

Eevolution of the equinoxes 189 

Revolution of the pole of the equator 

around the pole of the ecliptic 190 

Changes among the stars caused by 

precession 190 

The present pole-star not always such. 190 
What will be the pole-star 13,000 years 

hence? 190 

Cause of the precession of the equi- 
noxes 191 

Explain how the cause operates 191 

Proportionate effect of the sun and 

moon in producing precession 192 

The law of compound rotations 192 

Tropical year defined 193 

How much shorter than the sidereal 

year 193 

Use of the precession of the equinoxes 

in chronology 193 

Nutation, defined 1 94 

Explain its operation 194 

Cause of nutation 194 

Aberration, defined 195 

Illustrated by a diagram 195 

Amount of aberration 195 

Effect on the places of the stars 195 

Motion of the Apsides — the fact 

stated 196 



Art. 

Direction of this motion Iy6 

Time of revolution of the line of ap- 
sides 196 

Present longitude of the perihelion. . . 196 

Cause of advance of apsides 196 

Anomaly defined 197 

Anomalistic year, its length 197 

Slow change in duration of the seasons. 198 
Mean and True Places of the Sun. . 199 

Mean Motion defined 199 

Illustrated by surveying a field 199 

Mean and true longitudedistinguished 199 

Equations defined 200 

Their object 200 

Mean and True Anomaly defined 200 

Equation of the center 200 

Explain from the figure 200 

Chapter V. — The Moon — Lunar Geogra- 
phy — Phases of the Moon — Her Revo- 
lutions. 

Distance of the moon from the earth. 201 

Her mean horizontal parallax 201 

Her diameter 201 

Volume, density, and mass 201 

Shines by reflected light 202 

Appearance in the telescope 202 

Terminator defined 203 

Its appearance 203 

Proofs of mountains and valleys. .... 20D 

Form of the valleys 204 

Ring-mountains— bulwark plains 204 

Lava-lines seen at full moon 205 

Water, clouds, vegetation 205 

Explain the method of estimating the 

height of lunar mountains 206 

Has the moon an atmosphere? 209 

Improbability of identifying artificial 

structures in the moon 210 

Phases of the Moon, their cause... 211 
Successive appearances of the moon 

from one new moon to another 211 

Syzygies defined 211 

Explain the phases of the moon from 

figure 46 212 

Revolutions of the Moon. Period 

of her revolutions about the earth.. 213 

Her apparent orbit a great circle 213 

A sidereal month defined 213 

A synodical do. 213 

Length of each 213 

Why the synodical is longer 213 

How each is obtained 213 

Inclination of the lunar orbit 214 

Nodes defined 214 

Why the moon sometimes runs high 

and sometimes low 215 

Harvest moon defined 216 

Ditto explained 216 

Explain why the moon is nearer to us 

when on the meridian than when 

near the horizon . 217 

Time of the moon's revolution on its 

axis 218 

How known 218 

Librations explained 219 

Diurnal libration 21 9 

Length of the lunar days 220 



ANALYSIS. 



XI 



Earth never seen on the opposite side 
of the moon ' 220 

Appearances of the earth to a specta- 
tor on the moon . . 220 

"Whv the earth would appear to remain 
fixed 220 

Path of the moon in space 221 

How much more is the moon attracted 
toward the sun than toward the 
earth? 222 

"When does the sun act as a disturbing 
force upon the moon ? 222 

"Why does not the moon abandon the 
earth at the conjunction ? 222 

The moon's orbit concave toward the 
sun 223 

Chapter VI. — Lunar Irregularities. 

Specify their general cause 225 

Unequal action of the sun upon the 

earth and moon 225 

Oblique action of earth and sun 225 

Gravity of the moon toward the earth 

at the syzygies 226 

Gravity at the quadratures 226 

Explain the disturbances in the 

moon's motions from figure 48 228 

Figure of the moon's orbit 230 

How its figure is ascertained 230 

Moon's greatest and least apparent 

diameters 230 

Her greatest and least distances from 

the earth 230 

Perigee and Apogee defined 230 

Eccentricities of the solar and lunar 

orbits compared 230 

Moon's nodes, their change of place. . 231 
Rate of this change per annum. ...... 231 

Period of their revolution 231 

Irregular curve described by the 

moon 232 

Cause of the retrograde motion of 

nodes 232 

Explain from figure 50 232 

Svnodical revolution of the node de- 
stined 233 

Its period 233 

The Saros explained 233 

The Metonic Cycle. 234 

Golden Number 234 

Revolution of the line of apsides 235 

Its period 235 

How the places of the perigee may be 

found 235 

Moon's anomaly defined. 235 

Cause of the revolution of the apsides, 236 
Amount of the equation of the center. 237 

Ejection defined 238 

Its cause explained 239 

Variation defined 240 

Its cause 240 

Annual Equation explained 241 

How these irregularities were first dis- 
covered 242 

How many equations are applied to 

the moon's motions ? 242 

Method of proceeding in finding the 

moon's place . » 242 



Successive degrees of accuracy at- 
tained . . 242 

Periodic and secular irregularities dis- 
tinguished 243 

Acceleration of the moon's mean mo- 
tion explained 243 

Its consequences 243 

Lunar inequalities of latitude and 
parallax 244 

Chapter VII.— Eclipses. 

Eclipse of the moon, when it happens. 245 
Eclipse of the sun, when it happens. 245 

When only can each occur 245 

Why an eclipse does not occur at 

every new and full moon 245 

Why eclipses happen at two opposite 

months 245 

Circumstances which affect the length 

of the earth's shadow 246 

Semi-angle of the cone of the earth's 

shadow, to what equal 247 

Length of the earth's shadow 248 

Its breadth where it eclipses the moon. 249 

Lunar ecliptic limit defined. 250 

Solar do. 250 

Amount of the lunar ecliptic limit. . . 251 

Appulse defined. 251 

Partial, total, central eclipse, each de- 
fined 251 

Penumbra defined 252 

Semi-angle of the moon's penumbra, 

to what equal „ 253 

Semi-angle of a section of the penum- 
bra where the moon crosses it 254 

Moon's horizontal parallax increased 

A' wh y - 255 

Why the moon is visible in a total 
eclipse 256 

Calculation of eclipses, general mode 
of proceeding 257 

To find the exact time of the begin- 
ning, end, duration, and magnitude 
of a lunar eclipse, by figures 53, 54. . 258 

Elements of an eclipse defined 259 

Digits defined 260 

How the shadow of the moon travels 
over the eartli in a solar eclipse. . . . 262 

Why the calculation of a solar eclipse 
is more complicated than a lunar. . 262 

Velocity of the moon's shadow 263 

Different ways in which the shadow 
traverses the earth, according as 
the conjunction is near the node or 
near the limit 263 

When do the greatest eclipses happen ? 264 

Case in which the moon's shadow 
nearly reaches the earth 265 

How far may the shadow reach beyond 
the center of the earth ? . . . . . 265 

Greatest diameter of the moon's shad- 
ow where it traverses the earth. . . . 266 

Greatest portion of the earth's surface 
ever covered by the moon's penum- 
bra 267 

Moon's apparent diameter compared 
with the sun's 268 

Annular eclipse, its cause 26S 



Xll. 



ANALYSIS. 



Art. 
Direction in •which the eclipse passes 

on the sun's disk 269 

Greatest duration of total darkness.. 269 
Eclipses of the sun more frequent 

than of the moon, why ? 269 

Lunar eclipses oftener visible, why ?. 269 
Eadiation of light in a total eclipse of 

the sun . .... 269 

Interesting phenomena of a total 

eclipse of the sun 271 

Baily's Beads 271 

Flame-colored projections. . . . 271 

Chapter VI! I. — Longitude — Tides. 

Objects of the ancients in studying 

astronomy 272 

Ditto of the moderns 272 

Longitude. — How to find the differ- 
ence of longitude between two 

places 273 

Method by the Chronometer explained 274 
How to set the chronometer to Green- 
wich time, . 274 

Accuracy of some chronometers 274 

Objections to them 274 

Longitude by eclipses explained 275 

Lunar method of finding the longi- 
tude 276 

Circumstances which render this 

method somewhat difficult 277 

Difference of longitude accurately ob- 
tained by magnetic telegraph 278 

Tides, — defined. 279 

High, Low, Spring, Neap, Flood, and 

Ebb Tide, severally defined 279 

Similar tides on opposite sides of the 

earth 279 

Interval between two successive high 

tides 279 

Average height for the whole globe.. 279 

Extreme height 279 

Cause of the tides 280 

Explain by figure 56 . . , . 281 

Tide- wave defined 281 

Comparative effects of the sun and 

moon in raising the tide. . „ 282 

"Why the moon raises a higher tide 

than the sun 282 

Spring tides accounted for 283 

Neap tides, ditto 283 

Power of the sun or moon to raise the 

tide, in what ratio to its distance,. . 284 
Influence of the declinations of the 

sun and moon on the tides 2S5 

Explain from figures 57 and 58 235 

Motion of the tide-wave not progres- 
sive 286 

Tides of rivers, narrow bays, how 

produced . 2S7 

Cotidal Lines defined 287 

Derivative and Primitive tides distin- 
guished 287 

Velocity of the tide-wave, circum- 
stances which affect it 288 

Explain by figure 59 2^9 

Examples of very high tides .... 289 

Unit of altitude defined 290 

Unit of altitude for different places. . 290 



Art. 

Establishment of a port. ... ..... 291 

Tides on the coast of North America, 

whence derived . 292 

Why no tides in lakes and seas 293 

Intricacy of the problem of the tides. 294 
Atmospheric tide 295 

Chapter IX. — The Planets — Inferior: 
Planets — Mercury and Venus. 

Signification of the term planet 296 

Planets known from antiquity 296 

Planets added in 1781 and 1846 296 

Asteroids 296 

Primarv and Secondary Planets dis- 
tinguished 296 

Number of each 296 

Inclination of the planetary orbits to 

the ecliptic 297 

Inferior and Superior planets distin- 
guished .... . 295 

How the planets differ among them- 
selves 298 

Distances from the sun in miles 299 

Great dimensions of the planetary 

system 299 

Illustrated by the motion of a railway 

car 299 

Order by which the distances of the 

planets increase 299 

Bode's law of distances 299 

Mean distances, how determined ..... 299 

Diameters in miles 300 

Great diversity in respect to magni- 
tude „ 300 

How the real diameters are found from 

the apparent 300 

Periodic Times in months and years 301 
Which of the planets move rapidly 

and which slowly. 301 

Inferior Planets. — Proximity to the 

sun - 302 

Illustration by Fig. 60 302 

Conjunction defined — inferior and 

superior . . 303 

Synodical revolution denned 304 

How to find the synodical from the 

sidereal 304 

Motion of an inferior planet, when 

direct and when retrograde 305 

How these motions are affected by the 

earth's motions 305 

When the inferior planets are station- 
ary 306 

Elongation of the stationary points- 

for Mercury and Venus 306 

Phases of the inferior planets 307 

Relative distances from the sun 3f>8 

Eccentricity of their orbits. 309 

Mode of finding the period in time. . 310 
When is an inferior planet brightest? 311 
Diurnal revolutions of Mercury and 

Venus 312 

Venus as the morning and evening star 313 

Phenomena every eight years 314 

Transits of the Inferior Planets 

defined 315 

When they occur — why not at every 
inferior conjunction 315 



ANALYSIS. 



Xlll 



Art. 

Why those of Mercury in 'May and 

November • • • • 315 

Why those of Venus in June and De- 
cember 315 

Intervals between the transits of Mer- 
cury 31 ° 

Intervals between the transits of 

Venus 316 

How found 316 

Why so great an interest is attached 

to" the transits of Venus 317 

Why the sun's horizontal parallax 

cannot be found like the moon's.. 317 
Why distant places of observation 

are taken..... 318 

Process for the sun's hor. par. ex- 
plained from Fig. 63 318 

Sun's hor. par. in seconds 318 

To find the hor. par. of Venus and of 

Mars 318 

Atmosphere of Venus 319 

Satellites of Mercury and of Venus? 319 
Chapter X.— Superior Planets— Aster- 
oids— Motions of the Planets. 
Superior Planets, how distinguished 

from the Inferior 320 

Mars— size— distance from the sun . . 321 
Changes in apparent magnitude and 

brightness 321 

Phases of Mars, Fig. 64 322 

Telescopic appearances 323 

Satellite ?— ellipticity 323 

To find the hor. par. of Mars 324 

Asteroids— history of the first four. 325 
Distance from the sun— size— orbits . 326 

Modes of naming 327 

Jupiter— magnitude— figure— diurnal 

revolution 328 

Inclination of the axis to the orbit, 

and change of seasons 328 

Telescopic appearances 329 

Belts described and explained 330 

Satellites— how seen — names 331 

Magnitude— distances — periods 332 

Orbits — form — inclination 333 

Eclipses — their various phenomena, 

Fig. 65 334-335 

Shadows cast by the satellites on the 

Primary 336 

Longitude from the eclipses of Ju- 
piter's satellites 338-339 

Velocity of light, how discovered. . . 340 
Saturn — size — ring— telescopic view. 341 

King described 342 

Dimensions of the system 342 

Thinness of the rings 342 

Revolution of the ring around the 

sun 343-344 

Its changes and disappearances ex- 
plained 345-346 

Revolution of the ring in its own 

plane 347 

Satellites of Saturn — number and 

names s 348 

Eclipses 348 

Uranus— its discovery 340 

Size — periodic time — inclination 349 



Art. 

Satellites — number — peculiarities 350 

Neptune — distance — diameter — 

period 351 

History of its discovery 351-353 

Agreement of observation with theory 353 

Simultaneous discovery 353 

Results obtained by Walker 353 

Planetary Motions — two methods 

of studying them 354 

Appearances viewed from the sun. . . 355 

Motions of Mercury explained 355 

Form of orbit not seen from the sun. 355 
Why diagrams and orreries represent 

them erroneously 357 

Apparent motions of the planets 858 

Two causes make them unlike the 

real 358 

Apparent motions illustrated by 

Fig. 69 359 

Apparent motions of the Superior 

Planets 360 

Illustrated by Fig. 70 361 

Chapter XL — Determination of the 
Planetary Orbits— Kepler's Discov- 
eries — Elements of the Orbits of the 
Planets — Masses. 

Figure of the planetary orbits — an- 
cient ideas 362 

Notions of Ptolenry anil Hipparchus. 362 
Kepler — Investigation of the motions 

of Mars 363 

Discovery of the first law — the second 

—the third 363-365 

Modification of the third law 366 

Elements of the Planetary Orbits 

— enumerated 367 

Why not found like the lunar and 

solar orbits 368 

First steps of the process for finding 

the elements 369 

To convert geocentric longitudes and 

latitudes into heliocentric, Fig. 71. . 369 
To determine the position of the 

nodes 371 

To determine the inclination 371 

To find the periodic time. . 372 

The position of a planet which is 
most favorable for finding the ele- 
ments 373 

Exemplified in finding the periodic 

time of Saturn 374 

To determine the distance from the 

sun 375 

How the mean distance is found 375 

How the distance at any point in the 

orbit 375 

Method for the Inferior Planets 375 

Method for the Superior, Fig. 73 375 

To determine the place of ike perihe- 
lion 376 

To determine the epoch of passing the 

perihelion 376 

To find the eccentricity 377 

Quantity of Matter in the Sun and 

Planets 37$ 

How found in terms of the distances 
and periodic times 379 



XIV 



ANALYSIS. 



Art. 

How found by the spaces fallen 
through, Fig. 75 379 

How found in planets which have no 
satellites 380 

Densities, how found 381 

Specific gravities of the sun and plan- 
ets respectively 381 

Comparative densities 381 

Chapter XII. — Perturbations of Planets 
— Stability of the System — Numeri- 
cal Kelations — Problems. 

Perturbations — Numerous causes. . . 382 
Periodical and secular perturbations 

distinguished 382 

Case where the only bodies are a cen- 
tral and a revolving body 383 

How these irregularities have been 

discovered 383 

Minuteness of some perturbations. . . 383 
Whether the perturbations accumu- 
late indefinitely 384 

Stability of the system— how muin- . 

tained 384 

Nature of the evidence to prove the 

stability 384 

Invariability of the major axes 384 

Limits to the variation of the eccen- 
tricity 385 

Also to that of the inclination 385 

"What kind of perturbations are cu- 
mulative and what are oscillatory.. 385 
Conditions essential to this stability. 386 
Long inequality of Jupiter and Sat- 
urn 386 

Also of the Earth and Venus 386 

Numerical Eelations of the Plan- 
etary System • 387 

Change of velocity necessary on in- 
creasing the mass 387 

Also on increasing the distance 387 

Members of the solar system, how ad- 
justed 387 

Eelation between the rate of motion, 
distance, periodic time, and force of 

gravity 387 

.Demonstration of the rules 387 

The rules stated 388 

Given, the velocity, to find the other 

terms 388 

Given, the distance 388 

Given, the periodic time 3S8 

Given, the force of gravitation 388 

Eequired, the rate of motion, dis- 
tance, period, and force of gravita- 
tion respectively 388 

Problems 389 

Chapter XIIT. — Comets — Meteoric 
Showers. 

Comets — their several parts 390 

Number belonging to the system 391 

The six most remarkable 391 

Variations in magnitude and bright- 
ness 392 

To what owing 392 

Periods of revolution 393 

Distances from the sun 393 



Art. 

Figure of the orbit of Halley's comet. 393 

Source of the light 394 

Direction of the tails 394 

Quantity of matter in comets 395 

How the orbit of a comet may be 

changed 396 

Example in the comet of 1770 396 

Orbits and Motions of Comets 397 

How they differ from those of planets. 397 

Elements enumerated 398 

Their investigation, why difficult. 398-399 
How the return of a comet is predicted. 400 

Exemplified in Halley's comet 400 

Its return in 1759 and 1835 400-401 

Why an astronomical event of great 

interest 401 

Encke's comet — its period 402 

Question of a resisting medium. 402 

Comet of 1843 — its remarkable pecu- 
liarities 403 

Physical nature of comets 404 

Possibility of their striking the earth. 405 
Meteoric Showers— great shower of 

November, 1833 406 

Point of apparent radiation 406 

Extent and duration 406 

Periods of its recurrence 407 

Why an astronomical or cosmical 

phenomenon 407 

Of the periods of meteoric showers. . 407 
Conclusions respecting the meteors, 
as to their origin, nature, velocity, 

size, light, and heat 408 

Seasons for these conclusions 409 



Part III.— OF THE FIXED STAES 
AND SYSTEM OF THE WOELD. 

Chapter I. — Fixed Stars — Constella- 
tions. 

Why called fixed stars 410 

Classification 410 

Number in each class 410 

Antiquity of the constellations 411 

Their names — how individual stars 

are denoted 411 

Catalogues of the stars 412 

Number in the catalogue of Hippar- 

chus 412 

Number in Lalande's 412 

Utility of learning the constellations. 413 
Constellations of the Zodiac— Aries, 

Taurus 413 

Seven stars in Pleiades 413 

Gemini, Cancer 413 

Prsesepe, or the Bee-hive 413 

Leo, Virgo, Libra 413 

Scorpio, Sagittarius, Capricornus, 

Aquarius, Pisces 413 

Northern Constellations 414 

Ursa Minor, Ursa Major 414 

Draco 414 

Cepheus, Cassiopeia, Camelopard, 

Andromeda 415 

Perseus, Auriga, Leo Minor, Canes 

Venatici, Coma Berenices, Bootes.. 415 



ANALYSIS. 



XV 



Art. 

Corona Borealis, Hercules, Lyra, Cyg- 
nus 415 

Vulpecula, Aquila, Antinous, Del- 

phinus, Pegasus, Ophiuchus 415 

Southern Constellations , 416 

Orion, Lepus, Canis Major 416 

Canis Minor, Menoceros, Hydra 416 

Lesson for the middle of September. . 417 
Lesson for the middle of December.. . 417 

Lesson for the middle of March 417 

Lesson for the middle of June 417 

Chapter II. — Double Stars — Temporary 
Stars — Variable Stars — Clusters and 
Nebulae. 
Use of great telescopes in studying 

the stars 418 

Herschel's forty-feet telescope 419 

Eosse telescope 419 

Pulkova and Cambridge telescopes.. 419 

Double Stars — denned 420 

By whom discovered 421 

Examples — number 421 

"When merely optically double 421 

When physically double 421 

System of double, triple, and multiple 

stars 421 

Colors of the components 421 

Temporary Stars — defined 422 

Examples 422 

Variable Stars— denned 423 

Examples 423 

Evidence of activity among the stars. 423 

Clusters— examples 424 

Nebula— defined 424 

Examples — nebula of Andromeda 425 

Nebula of Hercules 425 

Magellanic clouds 425 

Nebula of Orion 425 

Use of great telescopes for these ob- 
jects 425 

Singular forms of nebulas 425 

Kesolvable and irresolvable distin- 

_ guished 426 

Signs of beauty and symmetry among 

the nebulas 426 

Nebulous Stars— defined "... 427 

Annular Nebula — defined 427 

Example in Lyra 427 

Planetary Nebula 427 

Resemblance to planets — great extent. 427 

Example in Andromeda 427 

MUky Way — cause of its peculiar 

light 428 

Number of its component stars 428 

Chapter III. — Motions of the Fixed 
Stars — Distances — Nature. 

Binary Stars— defined 429 

Number of these 429 

Periodic times — examples 429 

Law of gravitation among the stars. . 430 

Proper Motions of the stars 431 



Art 

Kesult on comparing the places of cer- 
tain stars in ancient and modern 

catalogues 431 

Motion of the solar system in space. . 432 

Point toward which it is moving. 432 

Eate of motion per annum 432 

Examples of great annual proper mo- 
tions • 433 

Distances of the Stars — how found. 434 
What is the base line for parallax?. . . 434 
Why it was supposed impossible to 

determine a parallax of less than \' r 434 
Distance implied by a parallax of \" '. 435 
Bessel's determination of the parallax 

of 61 Cygni 435 

His method of investigation 435 

Distance measured by the progress of 
light and by a railway car, respec- 
tively 435 

Actual period of revolution of the 

components of 61 Cygni 435 

Space described by the star annually. 435 
Reliance to be placed on Bessel's de- 
termination 435 

Nature of the Stars 436 

Size of Sirius compared with the sun. 436 
Proof that the fixed stars are suns. . . 437 

End for which they were made 438 

Arguments for a plurality of worlds. 438 

Chapter IV. — System of the "World. 

System of the world defined 439 

Complex character of early systems. . 439 

Things known to Pythagoras 440 

His visionary notions 440 

Rejection of his system 441 

Crystalline spheres of Eudoxus 441 

How the two motions were accounted 

for. 441 

Hipparchus — truths discovered by 

him 442 

Almagest of Ptolemy 442 

Ptolemaic System explained 444 

Illustrated by Fig. 81 445 

Defects of this system 446 

Objections to it 447 

Copernican System explained 449 

Arguments on which it rests 449 

Proofs that the planets revolve about 

the sun 449 

Proofs of systems among the stars. . . 450 
Exemplified in the Pleiades, Nebula 
of Hercules, Binary Stars, and Neb- 
ulas 450 

Uniformity of plan, in natural struc- 
tures 450 

Ascending orders of systems de- 
scribed 450 

Supposed center of the universe 450 

Central sun — where placed 450 

Reasons for believing that all the 
heavenly bodies are united in one 
grand system 451 



INTRODUCTION TO ASTRONOMY. 



Preliminary Observations. 

1. Astronomy is that science which treats of the heavenly 
bodies. 

More particularly, its object is to teach what is known 
respecting the Sun, Moon, Planets, Comets, and Fixed Stars ; 
and also to explain the methods by which this knowledge is 
acquired. Astronomy is sometimes divided into Descriptive, 
Physical, and Practical. Descriptive Astronomy respects 
facts', Physical Astronomy, causes / Practical Astronomy, 
the means of investigating the facts, whether by instruments, 
or by calculation. It is the province of Descriptive Astron 
omy to observe, classify, and record all the phenomena of the 
heavenly bodies, whether pertaining to those bodies individu- 
ally, or resulting from their motions and mutual relations. It 
is the part of Physical Astronomy to explain the causes of 
these phenomena, by investigating and applying the general 
laws on which they depend ; especially by tracing out all the 
consequences of the law of universal gravitation. Practical 
Astronomy lends its aid to both the other departments. 

2. Astronomy is the most ancient of all the sciences. At a 
period of very high antiquity, it was cultivated in Egypt, in 
Chaldea, in China, and in India. Such knowledge of the 
heavenly bodies as could be acquired by close and long-con- 
tinued observation, without the aid of instruments, was dili- 
gently amassed; and tables of the celestial motions were 
constructed, which could be used in predicting eclipses, and 
other astronomical phenomena. 

About 500 years before the Christian era, Pythagoras, of 
Greece, taught astronomy at the celebrated school at Crotona, 
and exhibited more correct views of the nature of the celestial 
motions than were entertained by any other astronomer of the 

l 



2 PRELIMINARY OBSERVATIONS. 

ancient world. His views, however, were not generally adopt- 
ed, but lay neglected for nearly 2000 years, when they were 
revived and established by Copernicus and Galileo. The most 
celebrated astronomical school of antiquity was at Alexandria, 
in Egypt, which was established and sustained by the Ptole- 
mies (Egyptian princes), about 300 years before the Christian 
era. The employment of instruments for measuring angles, 
and the introduction of trigonometrical calculations to aid the 
naked powers of observation, gave to the Alexandrian astrono- 
mers great advantages over all their predecessors. The most 
able astronomer of the Alexandrian school was Hipparchus, 
who was distinguished above all the ancients for the accuracy 
of his astronomical measurements and determinations. The 
knowledge of astronomy possessed by the Alexandrian school, 
and recorded in the Almagest, or great work of Ptolemy, con- 
stituted the chief of what was known of our science during the 
middle ages, until the fifteenth and sixteenth centuries, when 
the labors of Copernicus of Prussia, Tycho Brake of Denmark, 
Kepler of Germany, and Galileo of Italy, laid the solid foun- 
dations of modern astronomy. Copernicus expounded the true 
theory of the celestial motions ; Tycho Brahe carried the use 
of instruments and the art of astronomical observation to a far 
higher degree of accuracy than had ever been done before; 
Kepler discovered the great laws of the planetary motions; 
and Galileo, having first enjoyed the aid of the telescope, made 
innumerable discoveries in the solar system. ]STear the begin- 
ning of the eighteenth century, Sir Isaac Newton discovered, 
in the law of universal gravitation, the great principle that 
governs the celestial motions; and recently, La Place has 
more fully completed what Newton began, having followed 
out all the consequences of the law of universal gravitation, in 
his great work, the Mecanique Celeste. 

3. Among the ancients, astronomy was studied chiefly as 
subsidiary to astrology. Astrology was the art of divining 
future events by the stars. It was of two kinds, natural and 
judicial. Natural Astrology aimed at predicting remarkable 
occurrences in the natural world, as earthquakes, volcanoes, 
tempests, and pestilential diseases. Judicial Astrology aimed 
at foretelling the fates of individuals or of empires. 



PRELIMINARY OBSERVATIONS. 6 

4. Astronomers of every age have been distinguished for 
their persevering industry, and their great love of accuracy. 
They have uniformly aspired to an exactness in their inquiries 
far beyond what is aimed at in most geographical investiga- 
tions, satisfied with nothing short of numerical accuracy, 
wherever this is attainable ; and years of toilsome observation, 
or laborious calculation, have been spent with the hope of at- 
taining a few seconds nearer to the truth. Moreover, a severe 
but delightful labor is imposed on all who would arrive at a 
clear and satisfactory knowledge of the subject of astronomy. 
Diagrams, artificial globes, orreries, and familiar comparisons 
and illustrations, proposed by the author or the instructor, may 
afford essential aid to the learner, but nothing can convey to 
him a perfect comprehension of the celestial motions, without 
much diligent study and reflection. 

5. In expounding the doctrines of astronomy, we do not, as 
in geometry, claim that every thing shall be proved as soon as 
asserted. We may first put the learner in possession of the 
leading facts of the science, and afterwards explain to him the 
methods by which those facts were discovered, and by which 
they may be verified ; we may assume the principles of the 
true system of the world, and employ those principles in the 
explanation of many subordinate phenomena, while we reserve 
the discussion of the merits of the system itself, until the 
learner is extensively acquainted with astronomical facts, and 
therefore better able to appreciate the evidence by which the 
system is established. 

6. The Copernican System is that which is held to be the 
true system of the world. It maintains (1), That the apparent 
diurnal revolution of the heavenly bodies, from east to west, is 
owing to the real revolution of the earth on its own axis from 
west to east, in the same time ; and (2), That the sun is the 
center around which the earth and planets all revolve from 
west to east, contrary to the opinion that the earth is the center 
of motion of the sun and planets. 

7. We shall treat, first, of the Earth in its astronomical 
relations ; secondly, of the Solar System ; and, thirdly, of the 
Fixed Stars. 



PART I. -OF THE EARTH, 



CHAPTER I. 

OF THE FIGURE AND DIMENSIONS OF THE EARTH, AND THE DOCTRINE 
OF THE SPHERE. 



8. The figure of the earth is nearly globular. This fact is 
known, first, by the circular form of its shadow cast upon the 
moon in a lunar eclipse ; secondly, from analogy, each of the 
other planets being seen to be spherical ; thirdly, by our seeing 
the tops of distant objects while the other parts are invisible, as 
the topmast of a ship, while either leaving or approaching the 
shore, or the lantern of a light-house, which, when first descried 
at a distance at sea, appears to glimmer upon the very surface 
of the water ; fourthly, by the depression or dip of the horizon 
when the spectator is on an eminence ; and, finally, by actual 
observations and measurements, made for the express purpose 
of ascertaining the figure of the earth, by means of which 
astronomers are enabled to compute 
the distances from the center of the 
earth of various places on its surface, 
which distances are found to be nearly 
equal. 

9. The Dip of the Horizon, is the 
apparent angular depression of the 
horizon, to a spectator elevated above 
the general level of the earth. The 
eye thus situated takes in more than a 
celestial hemisphere, the excess being 
the measure of the dip. 

Thus, in Fig. 1, let AO represent 



Fig. 1. 




FIGURE AND DIMENSIONS. O 

the height of a mountain, ZO the direction of the plumb-line, 
HOB, a line passing through the station O, and at right angles 
to the plumb-line, C the center of the earth, DAE the portion 
of the earth's surface seen from O ; OD, OE, lines drawn from 
the place of the spectator to the most distant parts of the 
horizon, and CD a radius of the earth. The dip of the horizon 
is the angle HOD or ROE. Now the angle made between 
the direction of the plumb-line and that of the extreme line of 
the horizon or the surface of the sea, namely, the angle ZOD, 
can be easily measured ; and subtracting the right angle ZOH 
from ZOD, the remainder is the dip of the horizon, from which 
the length of the line OD may be calculated (see Art. 10), the 
height of the spectator, that is, the line OA, being known. 
This length, to whatever point of the horizon the line is drawn, 
is always found to be the same ; and hence it is inferred, that 
the boundary which limits the view on all sides is a circle. 
Moreover, at whatever elevation the dip of the horizon is taken, 
in any part of the earth, the space seen by the spectator is 
always circular. Hence the surface of the earth is spherical. 

1 0. The earth being a sphere, the dip of the horizon HOD 
= OCD. Therefore, to find the dip of the horizon correspond- 
ing to any given height AO* (the diameter of the earth being 
known), .we have in the triangle OCD, the right angle at D, 
and the two sides CD, CO, to find the angle OCD. Therefore, 

CO : rad. : : CD : cos. OCD. 

Learning the dip corresponding to different altitudes, by 
giving to the line AO different values, we may arrange the 
results in a table. 

* The learner wiil remark that the line AO, as drawn in the figure, is much 
larger in proportion to CA than is actually the case, and that the angle HOD is 
much too great for the reality. Such disproportions are very frequent in astro- 
nomical diagrams, especially when some of the parts are exceedingly small com- 
pared with others ; and hence the diagrams employed in astronomy are not to be 
regarded as true pictures of the magnitudes concerned, but merely as representing 
their abstract geometrical relations. 



6 

Table 



THK EARTH. 



the Dip of the Horizon at different elevations, 
from 1 foot to 100 feet.* " 



Feet. 


/ // 


Feet. 


/ // 


Feet. 


— , 


1 


0.59 


13 


3.33 


26 


5.01 


2 


1.24 


14 


3.41 


28 


5.13 


3 


1.42 


15 


3.49 


30 


5.23 


4 


1.58 


16 


3.56 


35 


5.49 


5 


2.12 


17 


4.03 


40 


6.14 


6 


2.25 


18 


4.11 


45 


6.36 


7 


2.36 


19 


4.17 


50 


6.58 


8 


2.41 


20 


4.24 


60 


7.37 


9 


2.57 


21 


4.31 


70 


8.14 


10 


3.07 


22 


4.37 


80 


8.48 


11 


3.16 


23 


4.43 


90 


9.20 


12 


3.25 


24 


4.49 


100 


9.51 



Such a table is of use in estimating the altitude of a body 
above the horizon, when the instrument (as usually happens) is 
more or less elevated above the general level of the earth. For 
if it is a star whose altitude above the horizon is required, the 
instrument being situated at O (Fig. 1), the inquiry is, how far 
the star is elevated above the level HOE, but the angle taken 
is that above the visible horizon OD. The dip, therefore, or 
the angle HOD, corresponding to the height of the point O, 
must be subtracted, to obtain the true altitude. On the Peak 
of Teneriffe, a mountain 13,000 feet high, Humboldt observed 
the surface of the sea to be depressed on all sides nearly 2 
degrees. The sun arose to him 12 minutes sooner than to an 
inhabitant of the plain ; and from the plain, the top of the 
mountain appeared enlightened 12 minutes before the rising or 
after the setting of the sun. 

1 1 . The foregoing considerations show that the form of the 
earth is spherical ; but more exact determinations prove that 
the earth, though nearly globular, is not exactly so : its diame- 
ter from the north to the south pole is about 26 miles less than 
through the equator, giving to the earth the form of an oblate 
spheroid,t or a flattened sphere resembling an orange. We 

* This table includes the allowance for refraction. 

f An oblate spheroid is the solid described by the revolution of an ellipse about 
its shorter axis. 



FIG UK K AND DIMENSIONS. 7 

shall reserve the explanations of the methods by which this 
fact is established, until the learner is better prepared than at 
present to understand them. 

1 2. The mean or average diameter of the earth is 7912.4 

miles, a measure which the learner should fix in his memory as a 

standard of comparison in astronomy, and of which he should 

endeavor to form the most adequate conception in his power. 

The circumference of the earth is about 25,000 miles (24857.5).* 

Although the surface of the earth is uneven, sometimes rising 

In high mountains, and sometimes descending in deep valleys, 

yet these elevations and depressions are so small in comparison 

with the immense volume of the globe, as hardly to occasion 

any sensible deviation from a surface uniformly curvilinear. 

The irregularities of the earth's surface in this view are no 

greater than the rough points on the rind of an orange, which 

do not perceptibly interrupt its continuity; for the highest 

mountain on the globe is only about five miles above the 

general level ; and the deepest mine hitherto opened is only 

5 1 

about half a mile.f Now — — = t^j, or about one-sixteen- 

hundredth part of the whole diameter, an inequality which, in 
an artificial globe of eighteen inches diameter, amounts to only 
the eighty-eighth part of an inch. 

1 3. The diameter of the earth, con- 
sidered as a perfect sphere, may be de- 
termined by means of observations on 
a mountain of known elevation, seen 
in the horizon from the sea. Let BD 
(Fig. 2), be a mountain of known 
height a, whose top is seen in the hori- 
zon by a spectator at A, h miles from it. 
Let x denote the radius of the earth. 
Then x 2 + h 2 = (x + a) 2 = x 2 +2ax+a 2 . 



* It will generally be sufficient to treasure up in the memory the round number, 
but sometimes, in astronomical calculations, the more exact number may be re- 
quired, and it is therefore inserted. 

f Sir John HerscheL 




8 THE EARTH. 

b 2 — a 2 
Hence, 2ax=o 2 —a 2 , and a?=— - — . For example, suppose the 

height of the mountain is just one mile; then it will be found, 
by observation, to be visible on the horizon at the distance of 

89 miles=5. Hence, ^=2?^==!?^=?=3960=radiuB 

2a 2 2 

of the earth, and 7920=the earth's diameter. 

1 4. Another method, and the most ancient, is to ascertain 
the distance on the surface of the earth, corresponding to a 
degree of latitude. Let us select two convenient places, one 
lying directly north of the other, whose difference of latitude 
is known. Suppose this difference to be 1° 30', and the dis- 
tance between the two places, as measured by a chain, to 
be 104 miles. Then, since there are 360 degrees of latitude 
in the entire circumference, 1° 30 ; : 104 :: 360° : 21960. And 

24960 -7944 
3l416~ ' ' * 

The foregoing approximations are sufficient to show that the 

earth is about 8,000 miles in diameter. 

15. The greatest difficulty in the way of acquiring correct 
views in astronomy, arises from the erroneous notions that pre- 
occupy the mind. To divest himself of these, the learner should 
conceive of the earth as a huge globe occupying a small portion 

Fig. 3. 




DOCTRINE OF THE SPHERE. 9 

of space, and encircled on all sides with the starry sphere. He 
should free his mind from its habitual proneness to consider 
one part of space as naturally up and another down, and 
view himself as subject to a force which binds him to the earth 
as truly as though he were fastened to it by some invisible 
cords or wires, as the needle attaches itself to all sides of a 
spherical loadstone. He should dwell on this point until it 
appears to him as truly up in the direction of BB, CC, DD 
(Fig. 3), when he is at B, C, and D, respectively, as in the 
direction of A A when he is at A. 

DOCTRINE OF THE SPHERE. 

16. The definitions of the different lines, points, and circles, 
which are used in astronomy, and the propositions founded 
upon them, compose the Doctrine of the Sphere* 

17. A section of a sphere'by a plane cutting it in any man- 
ner, is a circle. Great circles are those which pass through the 
center of the sphere, and divide it into two equal hemispheres : 
Small circles are such as do not pass through the center, but 
divide the sphere into two unequal parts. The circumference 
of every circle, whether great or small, is divided into 360 
equal parts called degrees. Hence, a degree is not a particular 
length of arc, but only a certain part of any whole circumference. 

1 8. The Axis of a circle is a straight line passing through 
its center at right angles to its plane. 

1 9. The Pole of a great circle is the point on the sphere where 
its axis cuts through the sphere. Every great circle has two 
poles, each of which is everywhere 90° from the great circle. 
For, the measure of an angle at the center of a sphere is the arc 
of a great circle intercepted between the two lines that con- 
tain the angle; and, since the angle made by the axis and any 
radius of the circle is a right angle, consequently its measure on 

s It is presumed that many of those who read this work will have studied 
Spherical Geometry ; but it is so important to the. student of astronomy to have 
a clear idea of the circles of the sphere, that it is thought best to introduce 
them here. , 



10 THE EARTH. 

the sphere, namely, the distance from the pole to the circum- 
ference of the circle, must be 90°. If two great circles cut each 
other at right angles, the poles of each circle lie in the circum- 
ference of the other circle. For each circle passes through the 
axis of the other. 

20. All great circles of the sphere cut each other in two 
points diametrically opposite, and consequently their points of 
section are 180° apart. For the line of common section is a 
diameter in both circles, and therefore bisects both. 

21. A great circle which passes through the pole of another 
great circle, cuts the latter at right angles. For, since it 
passes through the pole and the center of the circle, it must 
pass through the axis; which being at right angles to the 
plane of the circle, every plane which passes through it is at 
right angles to the same plane. 

The great circle which passes through the pole of another 
great circle, and is at right angles to it, is called a secondary 
to that circle. 

22. The angle made by two great circles on the surface of the 
sphere, is measured by the arc of another great circle, of which 
the angular point is the pole, being the arc of that great circle 
intercepted between those two circles. For this arc is the meas- 
ure of the angle formed at the center of the sphere by two radii, 
drawn at right angles to the line of common section of the two 
circles, one in one plane and the other in the other, which angle 
is therefore that of the inclination of those planes. 

23. In order to fix the position of any plane, either on the 
surface of the earth or in the heavens, both the earth and the 
heavens are conceived to be divided into separate portions by 
circles, which are imagined to cut through them in various ways. 
The earth, thus intersected, is called the terrestrial, and the 
heavens, the celestial sphere. The learner will remark that these 
circles have no existence in nature, but are mere landmarks, 
artificially contrived, for convenience of reference. On account 
of the immense distance of the heavenly bodies, they appear to 
us, wherever we are placed, to be fixed in the same concave 



DOCTRINE OF THE SPHERE. 11 

surface, or celestial vault. The great circles of the globe, ex- 
tended every way to meet the concave surface of the heavens, 
become circles of the celestial sphere. 

24. The Horizon is the great circle which divides the earth 
into upper and lower hemispheres, and separates the visible 
heavens from the invisible. This is the rational horizon. The 
sensible horizon is a circle touching the earth at the place of the 
spectator, and is bounded by the line in which the earth and 
skies seem to meet. The sensible horizon is parallel to the 
rational, but is distant from it by the semi-diameter of the earth, 
or nearly 4,000 miles. Still, so vast isthe distance of the starry 
sphere, that both these planes appear to cut that sphere in the 
same line ; so that we see the same hemisphere of stars that we 
should see if the upper half of the earth were removed, and we 
stood on the rational horizon. 

25. The poles of the horizon are the zenith and nadir. The 
Zenith is the point directly over our head, and the Nadir that 
directly under our feet. The plumb line is in the axis of the 
horizon, and consequently directed towards its poles. 

Every place on the surface of the earth has its own horizon ; 
and the traveller has a new horizon at every step, always ex- 
tending 90 degrees from his zenith in all directions. 

20. Vertical circles are those which pass through the poles 
of the horizon, perpendicular to it. 

The Meridian is that vertical circle which passes through 
the north and south points. 

The Prime Vertical is that vertical circle which passes 
through the east and west points. 

27. As in geometry we determine the position of any point 
by means of rectangular co-ordinates, or perpendiculars drawn 
from the point to planes at right angles to each other, so in 
astronomy we ascertain the place of a body, as a fixed star, by 
taking its angular distance from two great circles, one of which 
is perpendicular to the other. Thus the horizon and the 
meridian, or the horizon and the prime vertical, are co-ordinate 
circles used for such measurements. 



12 THE EARTH. 

The Altitude of a body is its elevation above the horizon 
measured on a vertical circle. 

The Azimuth of a body is its distance measured on the horizon 
from the meridian to a vertical circle passing through the body. 

The Amplitude of a body is its distance on the horizon from 
the prime vertical to a vertical circle passing through the body. 

Azimuth is reckoned 90° from either the north or south point ; 
and amplitude 90° from either the east or west point. Azimuth 
and amplitude are mutually complements of each other. When 
a point is on the horizon it is only necessary to count the num- 
ber of degrees of the horizon between that point and the 
meridian, in order to find its azimuth ; but if the point is above 
the horizon, then its azimuth is estimated by passing a vertical 
circle through it, and reckoning the azimuth from the point 
where this circle cuts the horizon. 

The Zenith Distance of a body is measured on a vertical cir- 
cle passing through that body. It is the complement of the 
altitude. 

28. The Axis of the Earth is the diameter on which the 
earth is conceived to turn in its diurnal revolution. The same 
line continued until it meets the starry concave, constitutes the 
axis of the celestial sphere. 

The Poles of the Earth are the extremities of the earth's axis : 
the Poles of the Heavens, the extremities of the celestial axis. 

29. The Equator is a great circle cutting the axis of the earth 
at right angles. Hence the axis of the earth is the axis of the 
equator, and its poles are the poles of the equator. The inter- 
section of the plane of the equator with the surface of the earth, 
constitutes the terrestrial, and with the concave sphere of 
the heavens, the celestial equator. The latter, by way of dis- 
tinction, is sometimes denominated the equinoctial. 

30. The secondaries to the equator, that is, the great circles 
passing through the poles of the equator, are called Merid- 
ians, because that secondary which passes through the zenith 
of any place is the meridian of that place, and is at right angles 
both to the equator and the horizon, passing as it does through 
the poles of both. (Art. 21.) These secondaries are also called 



DOCTRINE OF THE SPHERE. 13 

Hour Circles, because the arcs of trie equator intercepted be- 
tween them are used as measures of time. 

3 1 . The Latitude of a place on the earth is its distance from 
the equator north or south. The Polar Distance, or angular dis- 
tance from the nearest pole, is the complement of the latitude. 

32. The Longitude of a place is its distance from some stand- 
ard meridian, either east or west, measured on the equator. 
The meridian usually taken as the standard, is that of the Ob- 
servatory of Greenwich, near London. If a place is directly on 
the equator, we have only to inquire how many degrees of the 
equator there are between that place and the point where the 
meridian of Greenwich cuts the equator. If the place is north 
or south of the equator, then its longitude is the arc of the 
equator intercepted between the meridian which passes through 
the place, and the meridian of Greenwich. 

33. The Ecliptic is a great circle in which the earth performs 
its annual revolution around the sun. It passes through the 
center of the earth and the center of the sun. It is found by 
observation that the earth does not lie with its axis at right 
angles to the plane of the ecliptic, but that it is turned about 
23-J- degrees out of a perpendicular direction, making an angle 
with the plane itself of 66$°. The equator, therefore, must be 
turned the same distance out of a coincidence with the ecliptic, 
the two circles making an angle with each other of 23-J- 
(23° 27' 40"). It is particularly important for the learner to 
form correct ideas of the ecliptic, and of its relations to the 
equator, since to these two circles a great number of astro- 
nomical measurements and phenomena are referred. 

34. The Equinoctial Points, or Equinoxes,* are the inter- 
sections of the ecliptic and equator. The time when the sun 
crosses the equator in returning northward is called the vernal, 
and in going southward the autumnal equinox. The vernal 
equinox occurs about the 21st of March, and the autumnal the 
22d of September. 



* The term Equinoxes strictly denotes the times when the sun arrives at the 
equinoctial points, but it is also frequently used to denote those points themselves. 



14: THE EARTH. 

35. The Solstitial Points are the two points of the ecliptic 
most distant from the equator. The times when the sun comes 
to them are called solstices. The summer solstice occurs about 
the 22cl of June, and the winter solstice about the 22d of 
December. 

The ecliptic is divided into twelve equal parts of 30° each, 
called signs, which, beginning at the vernal equinox, succeed 
each other in the following order : 

Northern. Southern. 

1. Aries T 7. Libra ^ 

2. Taurus 8 8. Scorpio *n 

3. Gemini II 9. Sagittarius t 

4. Cancer S 10. Capricornus V3 

5. Leo & 11. Aquarius ™ 

6. Yirgo «K 12. Pisces X 

The mode of reckoning on the ecliptic, is by signs, degrees, 
minutes, and seconds. The sign is denoted either by its name 
or its number. Thus 100° may be expressed either as the 10th 
degree of Cancer, or as 3 s 10°. 

36. Of the various meridians, two are distinguished by the 
name of Coheres. The Equinoctial Cohere is the meridian which 
passes through the equinoctial points. The Solstitial Cohere is 
the meridian which passes through the solstitial points. As the 
solstitial points are 90° from the equinoctial points, so the sol- 
stitial colure is 90° from the equinoctial colure. It is also at 
right angles, or a secondary to both the ecliptic and equator. 
For, like every other meridian, it is of course perpendicular to 
the equator, passing through its poles. Moreover, the equinox, 
being a point both in the equator and in the ecliptic, is 90° 
from the solstice, from the pole of the equator, and from the 
pole of the ecliptic. Hence the solstitial colure, which passes 
through the solstice and the pole of the equator, passes also 
through the pole of the ecliptic, being the great circle of which 
the equinox itself is the pole. Consequently the solstitial 
colure is a secondary to both the equator and the ecliptic. 
(See Arts. 19, 20, 21.) 

37. The position of a celestial body is referred to the equator 
by its right ascension and declination. (See Art. 27.) Right 



DOCTRINE OF THE SPHERE. 



15 



Ascension is the angular distance from the vernal equinox, meas- 
ured on the equator. If a star is situated on the equator, then 
its right ascension is the number of degrees of the equator be- 
tween the star and the vernal equinox. But if the star is north 
or south of the equator, then its right ascension is the arc of 
the equator intercepted between the vernal equinox and that 
secondary to the equator which passes through the star. 
Declination is the distance of a body from the equator, meas- 
ured on a secondary to the latter. Therefore, right ascension 
and declination correspond to terrestrial longitude and latitude ; 
right ascension being reckoned from the equinoctial colure, in 
the same manner as longitude is reckoned from the meridian 
of Greenwich. On the other hand, celestial longitude and lati- 
tude are referred, not to the equator, but to the ecliptic. 
Celestial Longitude, is the distance of a body from the vernal 
equinox reckoned on the ecliptic. Celestial Latitude, is the 
distance from the ecliptic measured on a secondary to the lat- 
ter. Or, more briefly, Longitude is distance on the ecliptic : 
Latitude, distance from the ecliptic. The North Polar Dis- 
tance of a star, is the complement of its declination. 

Fig. 4. 

38. Parallels of Latitude are z 

small circles parallel to the equator. 
They constantly diminish in size as 
we go from the equator to the pole, 
the radius being always equal to the 
cosine of the latitude. In fig. 4, let 
HO be the horizon, EQ the equator, 
PP the axis of the earth, Z1ST the 
prime vertical, and ZL a parallel of 
latitude of any place Z. Then ZE is 
the latitude (Art. 31), and ZP the complement of the latitude; 
but Zrc, the radius of the parallel of latitude ZL, is the sine of 
ZP, and therefore the cosine of the latitude. 




39. The Tropics are the parallels of latitude that pass 
through the solstices. The northern tropic is called the tropic 
of Cancer ; the southern, the tropic of Capricorn. 

40. The Polar Circles are the parallels of latitude that pass 



16 THE EARTH. 

through the poles of the ecliptic, at the distance of 23J degrees 
from the pole of the earth. (Art. 33.) 

41. The earth is divided into five zones. That portion of 
the earth which lies between the tropics, is called the Torrid 
Zone / that between the tropics and the polar circles, the 
Temperate Zones / and that between the polar circles and the 

poles, the Frigid Zones. 

42. The Zodiac is the part of the celestial sphere which lies 
about 8 degrees on* each side of the ecliptic. This portion of 
the heavens is thus marked off by itself, because the planets 
are never seen further from the ecliptic than this limit. 

43. The elevation of the pole is equal to the latitude of the 
place. 

The arc PE (Fig. 4.)=ZO .\ PO=ZE, which equals the lati- 
tude. 

44. The elevation of the equator is equal to the complement 
of the latitude. : 

ZH=90°. But ZE=Lat. .*. EH=90-Lat.=colatitude. 

45. The distance of any place from the pole {or the polar 
distance) equals the complement of the latitude. 

EP=90°. But EZ=Lat. .-. ZP=90-Lat.=colatitude. 



1 



CHAPTEE II. 

DIURNAL REVOLUTION — ARTIFICIAL GLOBES — ASTRONOMICAL 
PROBLEMS. 

46. The apparent diurnal revolution of the heavenly bodies 
from east to west, is owing to the actual revolution of the 
earth on its own axis from west to east. If we conceive of a 
radius of the earth's equator extended until it meets the con- 
cave sphere of the heavens, then, as the earth revolves, the ex- 



DIURNAL REVOLUTION. Vj 

tremity of this line would trace out a curve on the face of the 
sky, namely, the celestial equator. In curves parallel to this, 
called the circles of diurnal revolution, . the heavenly bodies 
actually appear to move, every star having its own peculiar 
circle. After the learner has first rendered familiar the real 
motions of the earth from west to east, he may then, without 
danger of misconception, adopt the common language, that all 
the heavenly bodies revolve around the earth once a day from 
east to west, in circles parallel to the equator and to each 
other. 

47. The time occupied by a star in passing from any point 
in the meridian until it comes round to the same point again, 
is called a sidereal day, and measures the period of the earth's 
revolution on its axis. If we watch the returns of 'the same 
star from day to day, we shall find the intervals exactly equal 
to one another ; that is, the sidereal days are all equal* 
Whatever star we select for the observation, the same result 
will be obtained. The stars, therefore, always keep the same 
relative position, and have a common movement round the 
earth, — a consequence that naturally flows from,the hypothesis 
that their apparent motion is all produced by a single real 
motion, namely, that of the earth. The sun, moon, and 
planets, revolve in like manner, but their returns to the merid- 
ian are not, like those of the fixed stars, at exactly equal in- 
tervals. 

48. The appearances of the diurnal motions of the heavenly 
bodies are different in different parts of the earth, since every 
place has its own horizon (Art. 15), and different horizons are 
variously inclined to each other. Let us suppose the spectator 
viewing the diurnal revolutions, successively, from several dif- 
ferent positions on the earth. 

49. If he is on the equator, his horizon passes through both 
poles ; for the horizon cuts the celestial vault at 90 degrees in 
every direction from the zenith of the spectator ; but the pole 
is likewise 90 degrees from his zenith, and consequently, the 

• Allowance is here supposed to be made for the effects of precession, &c. 

2 



18 THE EARTH. 

pole must be in his horizon. The celestial equator coincides 
with his prime vertical, being a great circle passing through 
the east and west points. Since all the diurnal circles are par- 
allel to the equator, they are all, like the equator, perpendicu- 
lar to his horizon. Such a view of the heavenly bodies is 
called a right sphere ; or, 

A Eight Sphere is one in which all the daily revolutions of 
the heavenly bodies are in circles perpendicular to the horizon. 

A right sphere is seen only at the equator. Any star situ- 
ated in the celestial equator, would appear to rise directly in 
the east, to pass the meridian in the zenith of the spectator, 
and to set directly in the west. In proportion as stars are at a 
greater distance from the equator towards the pole, they de- 
scribe smaller and smaller circles, until, near the pole, their 
motion is hardly perceptible. In a right sphere every star re- 
mains an equal time above and below the horizon ; and since 
the times of their revolutions are equal, the velocities are as 
the lengths of the circles they describe. Consequently, as the 
stars are more remote from the equator towards the pole, their 
motions become slower, until, at the pole, if a star were there, 
it would appear stationary. 

50. If the spectator advances one degree towards the north 
pole, his horizon reaches one degree beyond the pole of the 
earth, and cuts the starry sphere one degree below the pole of 
the heavens, or below the north star if that be taken as the 
place of the pole. As he moves onward towards the pole, his 
horizon continually reaches further and further beyond it, until, 
when he comes to the pole of the earth, and under the pole of 
the heavens, his horizon reaches on all sides to the equator, and 
coincides with it. Moreover, since all the circles of daily mo- 
tion are parallel to the equator, they become, to the spectator 
-at the pole, parallel to the horizon. This is what constitutes a 
parallel sphere. Or, 

A Parallel Sphere is that in which all the circles of daily 
motion are parallel to the horizon. 

5 1 . To render this view of the heavens familiar, the learner 
should follow round in his mind a number of separate stars, 
one near the horizon, one a few degrees above it, and a third 



DIURNAL REVOLUTION. 



19 



near the zenith. To one who stood upon the north pole, the 
stars of the northern hemisphere would all be perpetually in 
view when not obscured by clouds or lost in the sun's light, 
and none of those of the southern hemisphere would ever be 
seen. The sun would be constantly above the horizon for six 
months in the year, and the remaining six constantly out of 
sight. That is, at the pole, the days and nights are each six 
months long. The phenomena at the south pole are similar to 
those at the north. 



52. A perfect parallel sphere can never be seen except at 
one of the poles, — a point which has never been actually 
reached by man; yet the British disco very -ships penetrated 
within a few degrees of the north pole, and of course enjoyed 
the view of a sphere nearly parallel. 

53. As the circles of daily motion are parallel to the horizon 
of the pole, and perpendicular to that of the equator, so at all 
places between the two, the diurnal motions are oblique to the 
horizon. This aspect of the heavens constitutes an oblique 
sphere, which is thus defined : 

An Oblique Sphere is that in which the circles of daily 
motion are oblique to the horizon. 

Suppose for example the spectator is at the latitude of fifty 
degrees. His horizon reaches 50° beyond the pole of the 
earth, and gives the same apparent elevation to the pole of the 
heavens. It cuts the equator, and all the circles of daily mo- 
tion, at an angle of 40°, being 
always equal to the co-altitude 
of the pole. Thus, let HO (Fig. 
5), represent the horizon, EQ the 
equator, and PP' the axis of the 
earth. Also, 11, mm, &c, par- 
allels of latitude. Then the ho- 
rizon of a spectator at Z, in lati- 
tude 50° reaches to 50° beyond 
the pole (Art. 50) ; and the angle 
ECH, is 40°. As we advance 
still further north, the elevation 
of the diurnal circles grows less 




THE EARTH. 



and less, and consequently the motions of the heavenly bodies 
more and more oblique, until finally, at the pole, where the lati- 
tude is 90°, the angle of elevation of the equator vanishes, and 
the horizon and the equator coincide with each other, as before 
stated. 

54. The ciecle of perpetual apparition, is the boundary 
of that space around the elevated pole, where the stars never set. 
Its distance from the pole is equal to the latitude of the place. 
For, since the altitude of the pole is equal to the latitude, a 
star whose polar distance is just equal to the latitude, will, 
when at its lowest point, only just reach the horizon ; and all 
the stars nearer the pole than this will evidently not descend 
so far as the horizon. 

Thus, mm (Fig. 5), is the circle of perpetual apparition, be- 
tween which and the north pole, the stars never set, and its 
distance from the pole OP is evidently equal to the elevation 
of the pole, and of course to the latitude. 

55. In the opposite hemisphere, a similar part of the sphere 
adjacent to the depressed pole never rises. Hence, 

The circle of perpetual occtjltation, is the boundary of 
that space around the depressed pole, within which the stars 
never rise. Thus, m'm! (Fig. 5), is the circle of perpetual oc- 
cultation, between which and the south pole the stars never 
rise. 

56. In an oblique sphere, the horizon cuts the circles of 
daily motion unequally. Towards the elevated pole, more 
than half the circle is above the horizon, and a greater and 
greater portion as the distance from the equator is increased, 
until finally, within the circle of perpetual apparition, the 
whole circle is above the horizon. Just the opposite takes 
place in the hemisphere next the depressed pole. Accordingly, 
when the sun is in the equator, as the equator and horizon, like 
all other great circles of the sphere, bisect each other, the days 
and nights are equal all over the globe. But when the sun is 
north of the equator, our days become longer than our nights, 
but shorter when the sun is south of the equator. Moreover, 
the higher the latitude, the greater is the inequality in the 



DIURNAL REVOLUTION. 21 

lengths of the days and nights. All these points will be read- 
ily understood by inspecting figure 5. 

57. Most of the phenomena of the diurnal revolution can be 
explained, either on the supposition that the celestial sphere 
actually all turns around the earth once in 24 hours, or that 
this motion of the heavens is merely apparent, arising from 
the revolution of the earth on its axis in the opposite direc- 
tion, — a motion of which we are insensible, as we sometimes 
lose the consciousness of our own motion in a ship or a steam- 
boat, and observe all external objects to be receding from us 
with a common motion. Proofs entirely conclusive and sat- 
isfactory, establish the fact, that it is the earth and not the 
celestial sphere that turns ; but these proofs are drawn from 
various sources, and the student is not prepared to appreciate 
their value, or even to understand some of them, until he has 
made considerable proficiency in the study of astronomy, and 
become familiar with a great variety of astronomical phenom- 
ena. To such a period of our course of instruction we there- 
fore postpone the discussion of the hypothesis of the earth's 
rotation on its axis. 

58. While we retain the same place on the earth, the diur- 
nal revolution occasions no change in our horizon, but our 
horizon goes round as well as ourselves. Let us first take our 
station on the equator at sunrise; our horizon now passes 
through both the poles, and through the sun, which we are to 
conceive of as at a great distance from the earth, and there- 
fore as cut, not by the terrestrial but by the celestial horizon. 
As the earth turns, the horizon dips more and more below the 
sun, at the rate of 15 degrees for every hour; and, as in the 
case of the polar star (Art. 50), the sun appears to rise at the 
same rate. In six hours, therefore, it is depressed 90 degrees 
below the sun, which brings us directly under the sun, which, 
for our present purpose, we may consider as having all the 
while maintained the same fixed position in space. The earth 
continues to turn, and in six hours more, it completely reverses 
the position of our horizon, so that the western part of the 
horizon which at sunrise was diametrically opposite to the sun 
now cuts the sun, and soon afterwards it rises above the level 



22 THE EARTH. 

of the sun, and the sun sets. During the next twelve hours, 
the sun continues on the invisible side of the sphere, until the 
horizon returns to the position from which it started, and a 
new day begins. 

59. Let us next contemplate the similar phenomena at the 
poles. Here the horizon, coinciding as it does with the equa- 
tor, would cut the sun through its center, and the sun would 
appear to revolve along the surface of the sea, one half above 
and the other half below the horizon. This supposes the sun 
in its annual revolution to be at one of the equinoxes. When 
the sun is north of the equator, it revolves continually round 
in a path which, during a single revolution, appears parallel 
to the horizon, and it is constantly day ; and when the sun is 
south of the equator, it is, for the same reason, continual night. 

60. We have endeavored to conceive of the manner in 
which the apparent diurnal movements of the sun are really 
produced at two stations, namely, in the right sphere, and in 
the parallel sphere. These two cases being clearly understood, 
there will be little difficulty in applying a similar explanation 
to an oblique sphere. 

ARTIFICIAL GLOBES. 

6 1 . Artificial globes are of two kinds, terrestrial and celes- 
tial. The first exhibits a miniature representation of the 
earth ; the second, of the visible heavens ; and both show the 
various circles by which the two spheres are respectively trav- 
ersed. Since all globes are similar solid figures, a small 
globe, imagined to be situated at the center of the earth or of 
the celestial vault, may represent all the visible objects and 
artificial divisions of either sphere, and with great accuracy 
and just proportions, though on a scale greatly reduced. The 
study of artificial globes, therefore, cannot be too strongly 
recommended to the student of astronomy.* 

* It were desirable, indeed, that every student of the science should have the 
celestial globe at least, constantly before him. One of a small size, as eight or 
nine inches, will answer the purpose, although globes of these dimensions cannot 
usually be relied on for nice measurements. 



ARTIFICIAL GLOBES. 23 

62. An artificial globe is encompassed from north to south 
by a strong brass ring, to represent the meridian of the place. 
This ring is made fast to the two poles, and thus supports the 
globe, while it is itself supported in a vertical position by 
means of a frame, the ring being usually let into a socket in 
which it may be easily slid, so as to give any required eleva- 
tion to the pole. The brass meridian is graduated each way 
from the equator to the pole 90°, to measure degrees of latitude 
or declination, according as the distance from the equator 
refers to a point on the earth or in the heavens. The horizon 
is represented by a broad zone, made broad for the convenience 
of carrying on it a circle of azimuth, another of amplitude, and 
a wide space on which are delineated the signs of the ecliptic, 
and the sun's place for every day in the year; not because 
these points have any special connection with the horizon, but 
because this broad surface furnishes a convenient place for 
recording them. 

63. Hour Circles are represented on the terrestrial globe 
by great circles drawn through the pole of the equator ; but, 
on the celestial globe, corresponding circles pass through the 
poles of the ecliptic, constituting circles of celestial latitude 
(Art. 37), while the brass meridian, being a secondary to the 
equinoctial, becomes an hour circle of any star which, by 
turning the globe, is brought under it. 

64. The Sour Index is a small circle described around the 
pole of the equator, on which are marked the hours of the day. 
As this circle turns along with the globe, it makes a complete 
revolution in the same time with the equator ; or, for any less 
period, the same number of degrees of this circle and of the 
equator pass under the meridian. Hence the hour index 
measures arcs of right ascension. (Art. 37.) 

65. The Quadrant of Altitude is a flexible strip of brass, 
graduated into ninety equal parts, corresponding in length to 
degrees on the globe, so that when applied to the globe and 
bent so as closely to fit its surface, it measures the angular 
distance between any two points. When the zero, or the 
point where the graduation begins, is laid on the pole of any 



24: THE EARTH. 

great circle, the 90th degree will reach to the circumference of 
that circle, and being therefore a great circle passing through 
the pole of another great circle, it becomes a secondary to the 
latter. (Art. 21.) Thus the quadrant of altitude may be used 
as a secondary to any great circle on the sphere ; but it is used 
chiefly as a secondary to the horizon, the point marked 90° 
being screwed fast to the pole of the horizon, that is, the 
zenith, and the other end, marked 0, being slid along between 
the surface of the sphere and the wooden horizon. It thus 
becomes a vertical circle, on which to measure the altitude of 
any star through which it passes, or from which to measure 
the azimuth of the star, which is the arc of the horizon inter- 
cepted between the meridian and the quadrant of altitude 
passing through the star. (Art. 27.) 

66. To rectify the globe for any place, the north pole must 
be elevated to the latitude of the place (Art. 43) ; then the 
equator and all the diurnal circles will have their due inclina- 
tion in respect to the horizon ; and, on turning the globe (the 
celestial globe west and the terrestrial east), every point on 
either globe will revolve as the same point does in nature ; and 
the relative situations of all places will be the same as on the 
respective native spheres. 

PROBLEMS ON THE TERRESTRIAL GLOBE. 

67. To find the Latitude and Longitude of a place: Turn 
the globe so as to bring the place to the brass meridian; then 
the degree and minute on the meridian directly over the place 
will indicate its latitude, and the point of the equator under 
the meridian, will show its longitude. 

Ex. What are the Latitude and Longitude of the citv of JSTew 
York? 

68. To find a place having its latitude and longitude given : 
Bring to the brass meridian the point of the equator correspond- 
ing to the longitude, and then at the degree of the meridian 
denoting the latitude, the place will be found. 

Ex. "What place on the globe is in Latitude 39 1ST. and Lon- 
gitude 77 W. \ 



PROBLEMS ON THE TERRESTRIAL GLOBE. 25 

69. To find the hearing and distance of two places: Rectify 
the globe for one of the places (Art. 66) ; screw the quadrant 
of altitude to the zenith,* and let it pass through the other 
place. Then the azimuth will give the bearing of the second 
place from the first, and the number of degrees on the quadrant 
of altitude, multiplied by 69i (the number of miles in a degree), 
will give the distance between the two places. 

Ex. What is the bearing of New Orleans from New York, 
and what is the distance between these places ? 

7 O. To determine the difference of time in different places : 
Bring the place that lies eastward of the other to the meridian, 
and set the hour index at XII. Turn the globe eastward until 
the other place comes to the meridian, then the index will show 
the hour at the second place when it is noon at the first. 

Ex. When it is noon at New York, what time is it at London ? 

7 1 . The hour being given at any place, to tell what hour it 
is in any other part of the world : Find the difference of time 
between the two places (Art. 70), and, if the place whose time 
is required is eastward of the other, add this difference to the 
given time, but if westward, subtract it. 

Ex. What time is it at Canton, in China, when it is 9 o'clock 
A. M. at New York? . 

72. To find the antoeci,\ the perioeci,% and the antipodes^ of 
any place : Bring the given place to the meridian ; then, under 
the meridian, in the opposite hemisphere, in the same degree 
of latitude, will be found the antoeci. The same place remain- 
ing under the meridian, set the index to XII, and turn the globe 
until the other XII is under the index ; then the perioeci will 
be on the meridian, under the same degree of latitude with the 
given place, and the antipodes will be under the meridian, in 
the same latitude, in the opposite hemisphere. 

Ex. Find the antoeci, the perioeci, and the antipodes of the 
citizens of New York. 

The antoeci have the same hour of the day, but different 



«■ The zenith will of course be the point of the rnevidaan over the place. 
+ avn oiKog. % Kept qikos. § avn ttsj. 



26 THE EARTH. 

seasons of the year ; the periceci have the same seasons, but 
opposite hours; and the antipodes have both opposite hours 
and opposite seasons. 

73. To rectify the globe for the sun's place : On the wooden, 
horizon, find the day of the month, and against it is given the 
sun's place in the ecliptic, expressed by signs and degrees.* 
Look for the same sign and degree on the ecliptic, bring that 
point to the meridian, and set the hour index to XII. To all 
places under the meridian it will then be noon. 

Ex. Rectify the globe for the sun's place on the 1st of Sep- 
tember. 

7 4. The latitude of the place being given, to find the time of 
the sun's rising and setting on any given day at that place : 
Having rectified the globe for the latitude (Art. 66), bring the 
sun's place in the ecliptic to the graduated edge of the meridian, 
and set the hour index to XII ; then turn the globe so as to 
bring the sun to the eastern and then to the western horizon, 
and the hour index will show the times of rising and setting 
respectively. 

Ex. At what time will the sun rise and set at New Haven, 
Lat. 41° 18', on the 10th of July? 

PROBLEMS ON THE CELESTIAL GLOBE. 

7 5. To find the Declination and Right Ascension of a heav- 
enly body : Bring the place of the body (whether the sun or a 
star) to the meridian. Then the degree and minute standing 
over it will show its declination, and the point of the equinoc- 
tial under the meridian will give its right ascension. It will 
be remarked, that the declination and right ascension are found 
in the same manner as latitude and longitude on the terrestrial 
globe. Right ascension is expressed either in degrees or in 
hours ; both being reckoned from the vernal equinox (Art. 37). 

Ex. "What is the declination and right ascension of the bright 
star Lyra ? — also of the sun on the 5th of June ? 



* The larger globes have the day of the month marked against the correspond- 
ing sign on the ecliptic itself. 



PROBLEMS ON THE CELESTIAL GLOBE. 27 

76. To represent the appearance of the heavens at any time : 
Rectify the globe for the latitude, bring the sun's place in the 
ecliptic to the meridian, and set the hour index to XII. ; then 
turn the globe westward until the index points to the given 
hour, and the constellations would then have the same appear- 
ance to an eye situated at the center of the globe, as they have 
at that moment in the sky. 

Ex. Required the aspect of the stars at New Haven, Lat. 
41° 18', at 10 o'clock, on the evening of December 5th. 

77. To find the altitude and azimuth of any star : Rectify 
the globe for the latitude and the sun's place, and let the quad- 
rant of altitude be screwed to the zenith, and be made to pass 
through the star. The arc on the quadrant, from the horizon 
to the star, will denote its altitude, and the arc of the horizon 
from the meridian to the quadrant, will be its azimuth. 

Ex. What are the altitude and azimuth of Sirius (the bright- 
est of the fixed stars) on the 25th of December at 10 o'clock 
in the evening, in Lat. 41° ? 

78. To find the angular distance of two stars from each 
other : Apply the zero mark of the quadrant of altitude to one 
of the stars, and the point of the quadrant which falls on the 
other star, will show the angular distance between the two. 

Ex. What is the distance between the two largest stars of 
the Great Bear?* 

79. To find the sun's meridian altitude, the latitude and day 
of the month being given : Having rectified the globe for the 
latitude (Art. 66), bring the sun's place in the ecliptic to the 
meridian, and count the number of degrees and minutes be- 
tween that point of the meridian and the zenith. The comple- 
ment of this arc will be the sun's meridian altitude. 

Ex. What is the sun's meridian altitude at noon on the 1st 
of August, in Lat. 41° 18' ? 

'■' These two stars are sometimes called "the Pointers," from the line which 
passes through them being always nearly in the direction of the north star. The 
angular distance between them is about 5°, and may be learned as a standard 
for reference in estimating, by the eye, the distance between any two points on 
the celestial vault. 



CHAPTEK III. 

OF PARALLAX, REFRACTION, AND TWILIGHT. 

80. Parallax is the apparent change of place which bodies 
undergo by being viewed from different points. This change is 
called diurnal parallax, when one point is at the surface of the 




earth, and the other at the center. Thus (Fig. 6), an observer 
on the surface at A would see the moon F at p on the sky, while 
from the center C, it would appear at P ; or, if the moon were 
at E, H would be its apparent place as seen from the center, 
and h from the surface. The angle AEC, or AFC, is the diur- 
nal parallax ; and HA or Yp is the parallactic arc. It appears, 
therefore, that the diurnal parallax of a body is the angle at 
that body, subtended by the earth's radius. 

81. Since then a heavenly body is liable to be referred to 
different points on the celestial vault, when seen from different 
parts of the earth, and thus some confusion occasioned in the 
determination of points on the celestial sphere, astronomers 
have agreed to consider the true place of a celestial object to 
be that where it would appear if seen from the center of the 



PARALLAX. 



29 



earth. The doctrine of parallax teaches how to reduce obser- 
vations made at any place on the surface of the earth, to such 
as they would be if made from the center. 

82. The angle AEC is called the horizontal parallax, which 
may be thus denned. Horizontal parallax is the change of 
position which a celestial body, appearing in the horizon as 
seen from the surface of the earth, would assume if viewed 
from the earth's center. It is the angle subtended by the 
earth's radius, when viewed perpendicularly from the body. 
If we consider any one of the triangles represented in the 
figure, ACG for example, 

Sin AGO : Sin GAZ (= Sin GAC) : : AC : CG ; 

a . -> n Sin GAZ x AC Sin GAZ 

.-. bin Parallax = ~~ oo — - 

Hence the sine of the angle of parallax, or (since the angle 
of parallax is always very small)* the parallax itself varies as 
the .sine of the zenith distance of the hody directly, and the dis- 
tance of the body from the center of the earth inversely. Par- 
allax, therefore, increases as a body approaches the horizon 
(but increasing with the sines, it increases much slower than 
in the simple ratio of the distance from the zenith), and dimin- 
ishes, as the distance from the spectator increases. Again, 
since the parallax AGC is as the sine of the zenith distance, 
let P represent the horizontal parallax, and P 7 the parallax 
at any altitude ; then, 

P' 

P' : P : : sin zenith dist. : sin 90V. P=- ,— • 

sin zen. dist. 

Hence, the horizontal parallax of a body equals its parallax 
at any altitude, divided by the sine of its -distance from the 
zenith. 

83. From observations, therefore, on the parallax of a body 
at any elevation, we are enabled to find the angle subtended 

* The moon, on account of its nearness to the earth, has the greatest horizon- 
tal parallax of any of the heavenly bodies ; yet this is less than 1° (being 57'), 
while the greatest parallax of any of the planets does not exceed 30". The dif- 
ference in an arc of 1°, between the length of the arc and the sine, is only 0".18. 



30 



THE EARTH. 



by the semi-diameter of the earth as seen perpendicularly from 
the body. Or if the horizontal parallax is given, the parallax 
at any altitude may be found, for 

P'=Px sin zenith distance. 

Hence, in the zenith the parallax is nothing, and is at its 
maximum in the horizon. 

84. It is evident from the figure, that the effect of parallax 
upon the place of a celestial body is to depress it. Thus, in 
consequence of parallax, E is depressed by the arc HA ; F by 
the arc ~Pp ; G by the arc Rr ; while O sustains no change. 
Hence, in all calculations respecting the altitude of the sun, 
moon, or planets, the amount of parallax is to be added ; the 
stars, as we shall see hereafter, have no sensible parallax. As 
the depression which arises from parallax is in the direction of 
a vertical circle, a body, when on the meridian, has only a 
parallax in declination ; but in other situations, there is at the 
same time a parallax in declination and right ascension ; for 
its direction being oblique to the equinoctial, it can be resolved 
into two parts, one of which (the declination) is perpendicular, 
and the other (the right ascension) is parallel to the equinoc- 
tial. 

85. The mode of determining the horizontal parallax, is as 
follows : 

Let O, O' (Fig. 7), be two places on the earth, situated un- 
der the same meridian, at a 
great distance from each oth- 
er ; one place, for example, at 
the Cape of Good Hope, and 
the other in the north of Eu- 
rope. The latitude of each 
place being known, the arc of 
the meridian 00' is known, 
and the angle OCO' also is 
known. Let the celestial 
body M (the moon for exam- 
ple), be observed simultane- 
ously at O and O', and its 
zenith distance at each place 



Fig. 7. 




PARALLAX. 31 

accurately taken, namely, ZY and Z / Y / ; then the angles ZOM 
and Z'O'M, and of course their supplements COM, C0 7 M are 
found. Join 00', and then in the isosceles triangle OCO 7 , 
determine the angles O, O', and the side 00' ; subtract O'OC 
from MOC, and OO'C from MO'C, and then in the triangle 
MOO', we can calculate MO and MO 7 . Finally, in the trian- 
gle MOC, two known sides and their contained angle will 
furnish the angle OMC (which is the parallax at the station 
O), and the distance MC. To obtain the horizontal parallax 
P, we have (Art. 82), 

OMC 
Sin ZOY' 

On this principle, the horizontal parallax of the moon was 
determined by La Caille and La Lande, two French astrono- 
mers, one stationed at the Cape of Good Hope, the other at 
Berlin ; and in a similar way the parallax of Mars was ascer- 
tained, by observations made simultaneously at the Cape of 
Good Hope and at Stockholm. 

86. On account of the great distance of the sun, his hori- 
zontal parallax, which is only 8". 6, cannot be accurately as- 
certained by this method. It can, however, be determined by 
means of the transits of Yenus, a process to be described here- 
after. 

87. The determination of the horizontal parallax of a celes- 
tial body is an element of great importance, since it furnishes 
the means of estimating the distance of the body from the 
center of the earth. Thus, if the angle AEC (Fig. 6) be 
found, the radius of the earth AC being known, we have in 
the triangle AEC, right-angled at A, the side AC and all the 
angles, to find the hypotenuse CE, which is the distance of 
the moon from the center of the earth. 



REFRACTION. 

88. While parallax depresses the celestial bodies subject to 
it, refraction elevates them ; and it affects alike the most dis- 
tant as well as nearer bodies, being occasioned by the change 



OZ THE EARTH. 

of direction which light undergoes in passing through the 
atmosphere. Let us conceive of the atmosphere as made up of 
a great number of concentric strata, as AA, BB, CC, and DD 
(Fig. 8), increasing rapidly in density (as is known to be the 

Fig. 8. 




fact) in approaching near to the surface of the earth. Let S 
be a star, from which a ray of light 8a enters the atmosphere 
at a, where, being turned towards the radius of the convex 
surface, it would change its direction into the line ab, and 
again into he, and <?0, reaching the eye at O. Now, since an 
object always appears in the direction in which the light 
finally strikes the eye, the star would be seen in the direction 
of the last ray cO, and the star would apparently change its 
place, in consequence of refraction, from S to S', being 
elevated out of its true position. Moreover, since on account 
of the constant increase of density in descending through the 
atmosphere, the light would be continually turned out of its 
course more and more ; it would therefore move, not in the 
polygon represented in the figure, but in a corresponding 
curve, whose curvature is rapidly increased near the surface of 
the earth. 



89. When a body is in the zenith, since a ray of light from 
it enters the atmosphere at right angles to the refracting me- 
dium, it suffers no refraction. Consequently, the position of 
the heavenly bodies, when in the zenith, is not changed by 
refraction ; while, near the horizon, where a ray of light strikes 
the medium very obliquely, and traverses the atmosphere 



PARALLAX. 



S3 



through its densest part, the refraction is greatest. The fol- 
lowing numbers, taken at different altitudes, will show how 
rapidly refraction diminishes from the horizon upward. The 
amount of refraction at the horizon is 34/ 00". At different 
elevations it is as follows. 



Elevation. 


Refraction. 


Elevation. 


Refraction. 


0° 10' 


32' 00" 


30° 


1' 40" 


20 


30 00 


40 


1 09 


1 00 


24 25 


45 


58 


5 00 


10 00 


60 


33 


10 00 


5 20 


80 


10 


20 00 


2 39 


90 


00 



From this table it appears, that while refraction at the horizon 
is 34 minutes, at so small an elevation as only 10 minutes above 
the horizon it loses 2 minutes, more than the entire change 
from the elevation of 30° to the zenith. From the horizon to 
1° above, the refraction is diminished nearly 10 minutes. The 
amount at the horizon, at 45°, and at 90° respectively, is 34/, 58", 
and 0. In finding the altitude of a heavenly body, the effect 
of parallax must be added, but that of refraction subtracted. 

90. Let us now learn the method by which the amount of 
refraction at different elevations is ascertained. To take the 
simplest case, we will suppose ourselves in a high latitude, 
where some of the stars within the circle of perpetual apparition 
pass through the zenith of the place. We measure the distance 
of such a star from the pole when on the meridian above the 
pole, that is, in the zenith, where it is not at all affected by 
refraction, and again its distance from the pole in its lower 
culmination. "Were it not for refraction, these two polar dis- 
tances would be equal, since, in the diurnal revolution of a 
star, it is in fact always at the same distance from the pole ; 
but, on account of refraction, the lower distance will be less 
than the upper, and the difference between the two will show 
the amount of refraction at the lower culmination, the latitude 
of the place being known. 

Example. At Paris, latitude 48° 50', a star was observed to 
pass the meridian 6' north of the zenith, and consequently 

3 



34 



THE EARTH. 



41° 4/ from the pole.* It should have passed the meridian at 
the same distance below the pole, but the distance was found to 
be only 40° 57' 35". Hence, 41° 4 / ~40° 57' 35"=6' 25" is the 
refraction due to that altitude, that is, at the altitude of 
7° 46 / =(48° 50 / -41° 4'). By taking similar observations in 
various places situated in high latitudes, the amount of refrac- 
tion might be ascertained for a number of different altitudes, 
and thus the law of increase in refraction, as we proceed from 
the zenith towards the horizon, might be ascertained. 



9 1 . Another method of finding the refraction at different 
altitudes, is as follows. Take the altitude of the sun or a star, 
whose right ascension and declination are known, and note the 
time by the clock. Observe also when it crosses the meridian, 
and the difference of time between the two observations will 
give the hour angle ZPa? (Fig. 9). In this triangle ZPa? we 




also know PZ the co-latitude and Pa? the co-declination. Hence 
we can find the co-altitude Zx, and of course the true altitude. 
Compare the altitude thus found with that before determined 
by observation, and the difference will be the refraction due 
to the apparent altitude. 

Ex. On May 1, 1738, at 5h. 20m. in the morning, Oassini 

* For the polar distance of the place=90-48 8 50'=41° 10' ; and 41° 10-6' 
=41° 4'. 



REFE4CTI0N. 35 

observed the altitude of the sun's center at Paris to be 5° 0' 14", 
The latitude of Paris being 48° 50' 10", and the sun's declina- 
tion at that time being 15° 0' 25" : Required the refraction. 

By spherical trigonometry, Zx will be found equal to 85° 10' 
8" ; consequently, the true altitude was 4° 49' 52". Now to 
5° 0' 14", the apparent altitude, 9" must be added for parallax, 
and we have 5° 0' 23'', the apparent altitude corrected for 
parallax. Hence, 5° 0' 23"-4° 49' 52"=10' 31", the refraction 
at the apparent altitude 5° 0' 14 



// # 



92. By these and similar methods, we could easily deter- 
mine the refraction due to any elevation above the horizon, 
provided the refracting medium (the atmosphere) were always 
uniform. But this is not the fact : the refracting power of the 
atmosphere is altered by changes in density and temperature, f 
Hence, in delicate observations, it is necessary to take into the 
account the state of the barometer and of the thermometer, the 
influence of the variations of each having been very carefully 
investigated, and rules having been given accordingly. With 
every precaution to insure accuracy, on account of the variable 
character of the refracting medium, the tables are not con- 
sidered as entirely accurate to a greater distance from the 
zenith than 74° ; but almost all astronomical observations are 
made at a greater altitude than this. 

93. Since the whole amount of refraction near the horizon 
exceeds 33', and the diameters of the sun and moon are severally 
less than this, these luminaries are in view both before they 
have actually risen and after they have set, 

94. The rapid increase of refraction near the horizon is 
strikingly evinced by the oval figure which the sun assumes 
when near the horizon, and which is seen to the greatest ad- 
vantage when light clouds enable us to view the solar disk. 
Were all parts of the sun equally raised by refraction, there 

* Gregory's Ast., p. 65. 

f It is said that the effects of humidity are insensible ; for the most accurate 
experiments seem to prove that watery vapor diminishes the density of air in the 
same ratio as its own refractive power is greater than that of air. (New Encyc, 
Brit, iii., 762.) 



36 THE EARTH. 

would be no change of figure ; but since the lower side is more 
refracted than the upper, the effect is to shorten the vertical 
diameter, and thus to give the disk an oval form. This effect 
is particularly remarkable when the sun, at its rising or setting, 
is observed from the top of a mountain, or at an elevation near 
the sea-shore ; for in such situations the rays of light make a 
greater angle than ordinary with a perpendicular to the re- 
fracting medium, and the amount of refraction is proportion- 
ally greater. In some cases of this kind, the shortening of the 
vertical diameter of the sun has been observed to amount to 6', 
or about one-fifth of the whole. * 

95. The apparent enlargement of the sun and moon in the 
horizon, arises from an optical illusion. These bodies in fact 
are not seen under so great an angle when in the horizon, as 
when on the meridian, for they are nearer to us in the latter 
case than in the former. The distance of the sun is indeed so 
great that it makes very little difference in his apparent diam- 
eter, whether he is viewed in the horizon or on the meridian ; 
but with the moon the case is otherwise ; its angular diameter, 
when measured with instruments, is perceptibly larger at the 
time of its culmination. "Why then do the sun and moon 
appear so much larger when near the horizon ? It is owing to 
that general law, explained in optics, by which we judge of 
the magnitudes of distant objects, not merely by the angle 
they subtend at the eye, but also by our impressions respecting 
their distance, allowing, under a given angle, a greater mag- 
nitude as we imagine the distance of a body to be greater. 
Now, on account of the numerous objects usually in sight 
between us and the sun, when on the horizon, he appears much 
further removed from us than when on the meridian, and we 
assign to him a proportionally greater magnitude. If we view 
the sun, in the two positions, through smoked glass, no such 
difference of size is observed, for here no objects are seen but 
the sun himself. 



* In extreme cold weather, this shortening of the sun's vertical diameter 
sometimes exceeds this amount. 



TWILIGHT. 



37 



TWILIGHT. 

96. Twilight also is another phenomenon depending upon 
the agency of the earth's atmosphere. It is due partly to re- 
fraction and partly to reflection, but mostly the latter. While 
the sun is within 18° of the horizon, before it rises or after it 
sets, some portion of its light is conveyed to us by means of nu- 
merous reflections from the atmosphere. Let AB (Fig. 10) be 

Fig. 10. 




the horizon of the spectator at A, and let SS be a ray of light 
from the sun when it is two or three degrees below the horizon. 
Then to the observer at A, the segment of the atmosphere ABS 
would be illuminated. To a spectator at C, whose horizon was 
CD, the small segment SDa? would be the twilight ; while, at 
E, the twilight would disappear altogether. 

97. At the equator, where the circles of daily motion are 
perpendicular to the horizon, the sun descends through 18° in 
an hour and twelve minutes (f|=l|-h.), and the light of day 
therefore declines rapidly, and as rapidly advances after day- 
break in the morning. At the pole, a constant twilight is en- 
joyed while the sun is within 18° of the horizon, occupying 
nearly two-thirds of the half year when the direct light of the 
sun is withdrawn, so that the progress from continual day to 
constant night is exceedingly gradual. To the inhabitants of 
an oblique sphere, the twilight is longer in proportion as the 
place is nearer the elevated pole. 

98. Were it not for the power the atmosphere has of dis- 



38 THE EARTH. 

persing the solar light, and scattering it in various directions, 
no objects would be visible to us out of direct sunshine ; every 
shadow of a passing cloud would be pitchy darkness ; the stars 
would be visible all day, and every apartment into which the 
sun had not direct admission, would be involved in the obscu- 
rity of night. This scattering action of the atmosphere on the 
solar light, is greatly increased by the irregularity of tempera- 
ture caused by the sun, which throws the atmosphere into a 
constant state of undulation, and by thus bringing together 
masses of air of different temperatures, produces partial reflec- 
tions and refractions at their common boundaries, by which 
means much light is turned aside from the direct course, and 
diverted to the purposes of general illumination.* In the up- 
per regions of the atmosphere, as on the tops of very high 
mountains, where the air is too much rarefied to reflect much 
light, the sky assumes a black appearance, and stars become 
visible in the day-time. 



CHAPTER IT. 

OF TIME. 



99. Time is a measured portion of indefinite duration. 
Any event may be taken as a measure of time, which divides 

a portion of duration into equal parts ; as the pulsations of the 
wrist, the vibrations of a pendulum, or the passage of sand 
from one vessel into another, as in the hour-glass. 

100. The great standard of time is the period of the revolu- 
tion of the earth on its axis, which, by the most exact observa- 
tions, is found to be always the same. The time of the earth's 
revolution on its axis is called a sidereal day, and is determined 
by the revolution of a star from the instant it crosses the me- 
ridian, until it comes round to the meridian again. This inter- 
val being called a sidereal day, it is divided into 24 sidereal 
hours. Observations taken upon numerous stars, in different 

8 Herschel. 



TIME. 39 

ages of the world, show that they all perform their diurnal 
revolutions in the same time, and that their motion during any 
part of the revolution is perfectly uniform. 

101. Solar time is reckoned by the apparent revolution of 
the sun, from the meridian round to the same meridian again. 
Were the sun stationary in the heavens, like a fixed star, the 
time of its apparent revolution would be equal to the revolu- 
tion of the earth on its axis, and the solar and the sidereal days 
would be equal. But since the sun passes from west to east, 
through 360° in 365J days, it moves eastward nearly 1° a day 
(59' 8".3). While, therefore, the earth is turning round on its 
axis, the sun is moving in the same direction, so that when we 
have come round under the same celestial meridian from which 
we started, we do not find the sun there, but he has moved 
eastward nearly a degree, and the earth must perform so much 
more than one complete revolution, in order to come under 
the sun again. ISTow since a place on the earth gains 360° in 
24 hours, how long will it take to gain 1° % 

24 
360 : 24 : : 1 : WnQ=^ m nearly. 

Hence the solar day is about 4 minutes longer than the side- 
real ; and if we were to reckon the sidereal day 24 hours, we 
should reckon the solar day 24h. 4m. To suit the purposes of 
society at large, however, it is found most convenient to reckon 
the solar day 24 hours, and to throw the fraction into the side- 
real day. Then, 

24h. 4m. : 24 : : 24 : 23h. 56m. (23h. 56 m 4 s .09)=the length of 
a sidereal day. 

102. The solar days, however, do not always differ from 
the sidereal by precisely the same fraction, since the increments 
of right ascension (Art. 37), which measure the difference be- 
tween a sidereal and a solar day, are not equal to each other. 
Apparent time is time reckoned by the revolutions of the sun 
from the meridian to the meridian again. These intervals 
being unequal, of course the apparent solar days are unequal 
to each other. 

103. Mean time is time reckoned by the average length of 



40 THE EARTH. 

all the solar days throughout the year. This is the period 
which constitutes the civil day of 24 hours, beginning when 
the sun is on the lower meridian, namely, at 12 o'clock at 
night, and counted by 12 hours from the lower to the upper 
culmination, and from the upper to the lower. The astronom- 
ical day is the apparent solar day counted through the whole 
24 hours, instead of by periods of 12 hours each, and begins at 
noon. Thus 10th day and 14th hour of astronomical timej 
would be 11th day and 2d hour of civil time. 

104. Clocks are usually regulated so as to indicate mean 
solar time ; yet as this is an artificial period, not marked off, 
like the sidereal day, by any natural event, it is necessary to 
know how much is to be added to or subtracted from the ap- 
parent solar time, in order to give the corresponding mean 
time. The interval by which apparent time differs from mean 
time, is called the equation of time. If a clock were con- 
structed (as it may be) so as to keep exactly with the sun, going 
faster or slower according as the increments of right ascension 
were greater or smaller, and another clock were regulated to 
mean time, then the difference of the two clocks at any period, 
would be the equation of time for that moment. If the ap- 
parent clock were faster than the mean, then the equation of 
time must be subtracted ; but if the apparent clock were slower 
than the mean, then the equation of time must be added, to 
give the mean time. The two clocks- would differ most about 
the 3d of November, when the apparent time is 16-} ra greater 
than the mean (16 m 17 s ). But, since apparent time is some- 
times greater and sometimes less than mean time, the two, 
must obviously be sometimes equal to each other. This is in 
fact the case four times a year, namely, April 15th, June 15th, 
September 1st, and December 22d. These epochs, however, 
do not remain constant ; for, on account of the change in the 
position of the perihelion, or the point where the earth is near- 
est the sun (which shifts its place from west to east about 12" 
a year), the period when the sun's motions are most rapid, as 
well as that when they are slowest, will occur at different parts 
of the year. The change is indeed exceedingly small in a sin- 
gle year; but in the progress of ages, the time of year when the 
sun's motion in its orbit is most accelerated, will not | be, as 



TIME. 



41 



at present, on the first of January, but may fall on the first of 
March, June, or any other day of the year, and the amount of 
the equation of time is obviously affected by the sun's distance 
from its perihelion, since the sun moves most rapidly when 
nearest the perihelion, and slowest when furthest from that point. 

105. The inequality of the solar days depends on two 
causes, the unequal motion of the earth in its orbit, and the 
inclination of the equator to the ecliptic. 

First, on account of the eccentricity* of the earth's orbit, the 
earth actually moves faster from the autumnal to the vernal 
equinox, than from the vernal to the autumnal, the difference 
of the two periods being about eight days (Td. 17h. 17m.) 

Fig. 11. 




Thus, let AEB (Fig. 11) represent the earth's orbit, S being 
the place of the sun, A the 'perihelion, or nearest distance of 
the earth from the sun, B the aphelion, or greatest distance, 
and E, E', E", positions of the earth in different points of its 
orbit. The place of the earth among the signs is the part of 

* The exact figure of the earth's orbit will be more particularly shown here- 
after. All that the student requires to know, in order to understand the present 
subject, is, that the earth's orbit is an ellipse, and that the earth's real motion, 
and consequently the sun's apparent motion, is greater in proportion as the earth 
is nearer the sun. 



42 THE EARTH. 

the heavens to which it would be referred if seen from the 
sun ; and the place of the sun is the part of the heavens to 
which it is referred as seen from the earth. Thus, when the 
earth is at E, it is said to be in Aries ; and as it moves from E 
through E' to A, its path in the heavens is through Aries, 
Taurus, Gemini, &c. Meanwhile the sun takes its place suc- 
cessively in Libra, Scorpio, Sagittarius, &c. JSTow, as will be 
shown more fully hereafter, the earth moves faster when 
proceeding from Aries through its perihelion to Libra, than 
from Libra through its aphelion to Aries, and, consequently, 
describes the half of its apparent orbit in the heavens, T, S =£=, 
sooner than the half =£=. V3. T- The line of the apsides, that is, 
the major axis of the ellipse, is so situated at present, that the 
perihelion is in the sign Cancer, nearly 100° (99° 30' 5") from 
the vernal equinox. The earth passes through it about the 
first of January, and then its velocity is the greatest in the 
whole year, being always greater as the distance is less, the 
angular velocity being inversely as the square of the distance, 
as will be shown by and by. 

106. But differences of time are not reckoned on the eclip- 
tic, but on the equinoctial ; for the ecliptic being oblique to 
the meridian in the diurnal motion, and cutting it at different 
angles at different times, equal portions will not pass under 
the meridian in equal times, and therefore such portions could 
not be employed, as they are in the equinoctial, as measures 
of time. If, therefore, the sun moved uniformly in his orbit, so 
as to make the daily increments of longitude equal, still the 
corresponding arcs of right ascension, which determine the 
lengths of the solar day, would be unequal. Let us start from 
the equinox, from which both longitude and right ascension 
are reckoned, the former on the ecliptic, the latter on the 
equinoctial. Suppose the sun has described 70° of longitude ; 
then to ascertain the corresponding arc of right ascension, we 
let a meridian pass through the sun : the point where it cuts 
the equator gives the sun's right ascension. Now since the 
ecliptic makes an acute angle with the meridian, while the 
equinoctial makes a right angle with it, consequently the arc 
of longitude is greater than the arc of right ascension. The 
difference, however, grows constantly less and less as we 



TIME. 



43 



approach the tropic, as the angle made between the ecliptic 
and the meridian constantly increases, until, when we reach 
the tropic, the meridian is at right angles to both circles, and 
the longitude and right ascension each equals 90°, and they 
are of course equal to each other. Beyond this, from the 
tropic to the other equinox, the arc of the ecliptic intercepted 
between the meridian and the autumnal equinox being greater 
than the corresponding arc of the equinoctial, of course its 
supplement, which measures the longitude, is less than the 
supplement of the corresponding arc of the equator which 
measures the right ascension. At the autumnal equinox again, 
the right ascension and longitude become equal. In a similar 
manner we might show that the daily increments of longitude 
and right ascension are unequal. 

In order to illustrate the foregoing points, let T =^= (Fig. 12) 

Fig. 12. 




represent the equator, T T «& the ecliptic, and PSE, PS'E', 
two meridians meeting the sun in S and S'. Then in the 
triangle TES, the arc of longitude TS, is greater than tE, the 
corresponding arc of right ascension ; but towards the tropic 
the difference between the two arcs evidently grows less and 
less, until at T the arcs become equal, being each 90°. But, 
beyond the tropic, since TE'=a=, TS'=£=, are equal to each other, 
each being equal to 180°, and since S'=^ is greater than E'=^, 
therefore TS' must be less than TE'. 



44 THE EARTH. 

107. As the whole arc of right ascension reckoned from the 
first of Aries, does not keep uniform pace with the correspond- 
ing arc of longitude, so the daily increments of right ascension 
differ from those of longitude. If we suppose in the quadrant 
"ifT, points taken to mark the progress of the sun from day to 
day, and let meridians like PSE pass through these points, the 
arcs of the ecliptic embraced between the meridians will be the 
daily increments of longitude, while the corresponding parts 
of the equinoctial will be the daily increments of right ascen- 
sion. Near T, the oblique direction in which the ecliptic cuts 
the meridian, will make the daily increments of longitude 
exceed those of right ascension ; but this advantage is dimin- 
ished as we approach the tropic, where the ecliptic becomes 
less oblique, and finally parallel to the equinoctial ; while the 
convergence of the meridians contributes still further to lessen 
the ratios of the daily increments of longitude to those of right 
ascension. Hence, at first, the diurnal arcs of right ascension 
are less than those of longitude, but afterward greater ; and 
they continue greater for a similar distance beyond the tropic. 

108. From the foregoing considerations it appears, that the 
diurnal arcs of right ascension, by which the difference between 
the sidereal and the solar days is measured, are unequal, on 
account both of the unequal motion of the sun in his orbit, 
and of the inclination of his orbit to the equinoctial. 

109. As astronomical time commences when the sun is on 
the meridian, so sidereal time commences when the vernal 
equinox is on the meridian, and is also counted from to 24 
hours. By 3 o'clock, for instance, of sidereal time, we mean 
that it is three hours since the vernal equinox crossed the 
meridian ; as we say it is 3 o'clock of astronomical or of civil 
time, when it is three hours since the sun crossed the meridian. 



THE CALENDAR. 

HO. The astronomical year is the time in which the sun 
makes one revolution in the ecliptic, and consists of 365d. 5h. 
48m. 51\60. The civil year consists of 365 days. The differ- 
ence is nearly 6 hours, making one day in four years. 



THE CALENDAR. 45 

111. The most ancient nations determined the number of 
days in the year by means of the stylus, a perpendicular rod 
which cast its shadow on a smooth plane, bearing a meridian 
line. The time when the shadow was shortest, would indicate 
the day of the summer solstice; and the number of days 
which elapsed until the shadow returned to the same length 
again, would show the number of days in the year. This was 
found to be 365 whole days, and accordingly this period was 
adopted for the civil year. Such a difference, however, be- 
tween the civil and astronomical years, at length threw all 
dates into confusion. For, if at first the summer solstice hap- 
pened on the 21st of June, at the end of four years, the sun 
would not have reached the solstice until the 22d of June, 
that is, it would have been behind its time. At the end of 
the next four years the solstice would fall on the 23d ; and in 
process of time it would fall successively on every day of the 
year. The same would be true of any other fixed date. 
Julius Caesar made the first correction of the calendar, by in- 
troducing an intercalary day every fourth year, making Feb- 
ruary to consist of 29 instead of 28 days, and of course the 
whole year to consist of 366 days. This fourth year was de- 
nominated Bissextile.* It is also called Leap Year. 

112. But the true correction was not 6 hours, but 5h. 
49m. ; hence the intercalation was too great by 11 minutes. 
This small fraction would amount in 100 years to f of a day, 
and in 1000 years to more than 7 days. From the year 325 to 
1582, it had -in fact amounted to about 10 days; for it was 
known that in 325, the vernal equinox fell on the 21st of 
March, whereas, in 1582 it fell on the 11th. In order to re- 
store the equinox to the same date, Pope Gregory XIII. 
decreed that the year should be brought forward ten days, by 
reckoning the 5th of October the 15th. In order to prevent 
the calendar from falling into confusion afterward, the fol- 
lowing rule was adopted : 

Every year whose number is not divisible by 4 without a re- 
mainder, consists of 365 days / every year which is so divisible, 
but is not divisible by 100, of 366 ; every year divisible, by 100 

* The sextus dies ante Kalendas being reckoned twice (Bis). 



46 THE EARTH. 

but not by 400, again of 365 ; and every year divisible by 400, 
of 366. 

Thus the year 1838, not being divisible by four, contains 365 
days, while 1836 and 1840 are leap years. Yet to make every 
fourth year consist of 366 days would increase it too much by 
about £ of a day in 100 years ; therefore every hundredth year 
has only 365 days. Thus 1800, although divisible by 4, was 
not a leap year, but a common year. But we have allowed a 
whole day in a hundred years, whereas we ought to have al- 
lowed only three-fourths of a day. Hence, in 400 years we 
should allow a day too much, and therefore we let the 400th 
year remain a leap year. This rule involves an error of less 
than a day in 4237 years.* If the rule were extended by 
making every year divisible by 4,000 (which would now con- 
sist of 366 days) to consist of 365 days, the error would not be 
more than one day in 100,000 years.f 

113. This reformation of the calendar was not adopted in 
England until 1752, by which time the error in the Julian 
calendar amounted to about 11 days. The year was brought 
forward, by reckoning the 3d of September the 14th. Pre- 
vious to that time the year began the 25th of March ; but it was 
now made to begin on the 1st of January, thus shortening the 
preceding year, 1751, one quarter 4 

114. As in the year 1582, the error in the Julian calendar 
amounted to 10 days, and increased by f of a day in a cen- 
tury, at present the correction is 12 days ; and the number of 
the year will differ by one with respect to dates between the 
1st of January and the 25th of March. 

Examples. General Washington was born Feb. 11, 1731, 
old style ; to what date does this correspond in new style ? 

As the date is the earlier part of the 18th century, the cor- 
rection is 11 days, which makes the birthday fall on the 22d 



« Woodhouse, p. 874. f Herschel's Ast., p. 384. 

% Russia, and the Greek Church generally, adhere to the old style. In order 
to make the Russian dates correspond to ours, we must add to them 12 days. 
France and other Catholic countries, adopted the Gregorian calendar soon after 
it was promulgated. 



THE CALENDAR. 47 

of February ; and since the year 1731 closed the 25th of 
March, while according to new style 1732 would have com- 
menced on the preceding 1st of January ; therefore, the time 
required is Feb. 22d, 1732. It is usual, in such cases, to write 
both years, thus: Feb. 11, 1731-2, O. S. 

2. A great eclipse of the sun happened May 15th, 1836 ; to 
what date would this time correspond in old style? 

Ans. May 3d. 

115. The common year begins and ends on the same day of 
the week ; but leap year ends one day later in the week than it 
began. 

For 52x7=364 days; if, therefore, the year begins on Tues- 
day, for example, 364 days would complete 52 weeks, and one 
day would be left to begin another week, and the following 
year would begin on Wednesday. Hence, generally, any day 
of the month is one day later in the week than the same day 
of the preceding year. Thus, July 4th, 1861, falls on Thursday ; 
1862, on Friday ; 1863, on Saturday. But, in leap-year, this 
rule applies only till the end of February. From that time to 
the same date in the year following, every day of a month 
falls two days later in the week than during the previous year. 
Thus, July 4th, 1871, is Tuesday; 1872, Thursday; and Feb- 
ruary 2d, 1872, is Friday ; 1873, Sunday. 

116. Fortunately for astronomy, the confusion of dates in- 
volved in different calendars affects recorded observations but 
little. Kemarkable eclipses, for example, can be calculated 
back for several thousand years, without any danger of mis- 
taking the day of their occurrence ; and whenever any such 
eclipse is so interwoven with the account given by an ancient 
author of some historical event, as to indicate precisely the 
interval of time between the eclipse and the event, and at the 
same time completely to identify the eclipse, that date is re- 
covered and fixed forever.* 



* An elaborate view of the Calendar may be found in Delambre's Astronomy, 
t. III. A useful table for finding the day of the week of any given date, is in- 
serted in the American Almanac for 1832, p. 72. 



CHAPTER V 



OF ASTRONOMICAL INSTRUMENTS AND PROBLEMS FIGURE AND 

DENSITY OF THE EARTH. 

117. The most ancient astronomers employed no instru- 
ments for measuring angles, but acquired their knowledge of 
the heavenly bodies by long-continued and most attentive in- 
spection with the naked eye. In the Alexandrian school, 
about 300 years before the Christian era, instruments began 
to be freely used, and thenceforward trigonometry lent a pow- 
erful aid to the science of astronomy. Tycho Brahe, in the 
16th century, formed a new era in practical astronomy, and 
carried the measurement of angles to 10", — a degree of accu- 
racy truly wonderful, considering that he had not the advan- 
tage of the telescope. By the application of the telescope to 
astronomical instruments, a far better denned view of objects 
was acquired, and a far greater degree of refinement was at- 
tainable. The astronomers royal of Great Britain perfected 
the art of observation, bringing the measurement of angles to 
1", and the estimation of differences of time to -fo of a second. 
Beyond this degree of refinement it is supposed that we can- 
not advance, since unavoidable errors arising from the uncer- 
tainties of refraction, and the necessary imperfection of instru- 
ments, forbid us to hope for a more accurate determination 
than this. But a little reflection will show us, that 1" on the 
limb of an astronomical instrument, must be a space exceed- 
ingly small. Suppose the circle, on which the angle is meas- 

, , « x . ,. ^ 12x3.14159 , . , 

urea, be one loot in diameter. Then -— = T V mch= 

ooO 

space occupied by 1°. Hence =— =space of 1', and 

— -— = space of 1". Such minute angles can be measured 

only by large circles. If, for example, a circle is 20 feet in 
diameter, a degree on its periphery would occupy a space 20 



THE CALENDAR. 49 

times as large as a degree on a circle of 1 foot. A degree 
therefore of the limb of such an instrument would occupy a 
space of 2 inches ; one minute, ^ of an inch ; and one second, 
xsV o of an inch. 

118, But the actual divisions on the limb of an astronomi- 
cal instrument never extend to seconds : in the smaller instru- 
ments they reach only to 10', and on the largest rarely lower 
than V. The subdivision of these spaces is carried on by 
means of the Yernier, which may be thus defined : 

A Yernier is a contrivance attached to the graduated limb 
of an instrument, for the purpose of measuring aliquot parts 
of the smallest spaces into which the instrument is divided. 

The vernier is usually a narrow zone of metal, which is 
made to slide on the graduated limb. Its divisions correspond 
to those on the limb, except that they are a little larger,* one- 
tenth, for example, so that ten divisions on the vernier would 
equal eleven on the limb. Suppose now that our instrument 
is graduated to degrees only, but the altitude of a certain star 
is found to be 40° and a fraction, or 40° -fee. In order to 
estimate the amount of this fraction, we bring the zero-point 
of the vernier to coincide with the point which indicates the 
exact altitude, or 40°-f-». We then look along the vernier 
until we find where one of its divisions coincides with one of 
the divisions of the limb. Let this be at the fourth division of 
the vernier. In four divisions, therefore, the vernier has 
gained upon the divisions of the limb, a space equal to x ; and 
since, in the. case supposed, it gains ^ of a degree, or 6' at 
each division, the entire gain is 24', and the arc in question is 
40° 24'. 

119. As the vernier is used in the barometer, where its 
application is more easily seen than in astronomical instru- 
ments, while the principle is the same in both cases, let us see 
how it is applied to measure the exact height of a column of 
mercury. Let AB (Fig. 13) represent the upper part of a 
barometer, the level of the mercury being at 0, namely, at 30.3 
inches, and nearly another tenth. The vernier being brought 

« In the more modern instruments the divisions of the vernier are smaller 
than those of the limb. 

4 



50 



THE EARTH. 



(by a screw which is usually attached to it) to coincide with the 
surface of the mercury, we look along Fig. 13. 

down the scale, until we find that the 
coincidence is at the 8th division of 
the vernier. Now as the vernier gains 

ts °f To = iio °f an i ncn at eacn ^ivi- 
si on upward, it of course gains r f in 
eight divisions. The fractional quan- 
tity, therefore, is .08 of an inch, and 
the height of the mercury is 30.38. If 
the divisions of the vernier were such, 
that each gained ^\ (when 60 on the 
vernier would equal 61 on the limb) 
on a limb divided into degrees, we 
could at once take off minutes ; and 
were the limb graduated to minutes, 
we could in a similar way read off seconds. 



A 
C 

B 








- 


r— 31 
30 

—29 


1 


1 

2 
3 

4: 

5 
G 
7 
8 
8 
11) 


— 


11 







1 20. The instruments most used for astronomical observa- 
tions, are the Transit Instrument, the Astronomical Clock, the 
Mural Circle, and the Sextant. A large portion of all the 
obseiwations made in an astronomical observatory, are taken 
on the meridian. When a heavenly body is on the meridian, 
being at its highest point above the horizon, it is then least 
affected by refraction and parallax ; its zenith distance (from 
which its altitude and declination are easily derived) is readily 
estimated ; and its right ascension may be very conveniently 
and accurately determined by means of the astronomical clock. 
Having the right ascension and declination of a heavenly 
body, various other particulars resj)ecting its position may be 
found, as we shall see hereafter, by the aid of spherical trigo- 
nometry. Let us then first turn our attention to the instru- 
ments employed for determining the right ascension and decli- 
nation. They are the Transit Instrument, the Astronomical 
Clock, and the Mural Circle. 

121. The Transit Instrument is a telescope, which is fixed 
permanently in the meridian, and moves only in that plane. 
It rests on a horizontal axis, which consists of two hollow 
cones applied base to base, a form uniting lightness and 



ASTRONOMICAL INSTRUMENTS. 



51 



strength. The two ends of the axis rest on two firm supports, 
as pillars of stone, for example, usually built up from the 
ground, and so related to the building as to be as free as possible 
from all agitation. In figure 14, AD represents the telescope, 
E, "W, massive stone pillars supporting the horizontal axis, 
beneath which is seen a spirit-level (which is used to bring 
the axis to< a horizontal position), and n a vertical circle grad- 
uated into degrees and minutes. This circle serves the pur- 
pose of placing the instrument at any required altitude or dis- 
tance from the zenith, and of course for determining altitudes 
and zenith distances. 




122. Various methods are described in works on practical 
astronomy, for placing the Transit Instrument accurately in the 
meridian. The following method, by observations on the pole- 
star, may serve as an example. If the instrument be directed 
towards the north star, and so adjusted that the star Alioth (the 
first in the tail of the Great Bear) and the pole-star are both in 
the same vertical circle, the former below the pole and the lat- 
ter above it, the instrument will be nearly in the plane of the 
meridian. To adjust it more exactly, compare the time oeeu 



52 



THE EARTH. 



pied by the pole-star in passing from its upper to its lower 
culmination, with the time of passing from its lower to its upper 
culmination. These two intervals ought to be precisely equal ; 
and if they are so, the instrument is truly placed in the merid- 
ian ; but if they are not equal, the position of the instrument 
must be shifted until they become exactly equal. 



1 23. The line of collimation of a telescope, is a line joining 
the center of the object-glass with the center of the eye-glass. 
When the transit instrument is properly adjusted, this line, as 
the instrument is turned on its axis, moves in the plane of the 
meridian. Having, by means of the vertical circle w, set the 
instrument at the known altitude or zenith distance of any star, 
upon which we wish to make observations, we wait until the 
star enters the field of the tel- 
escope, and note the exact in- 
stant when it crosses the ver- 
tical wire in the center of the 
field, which wire marks the 
true plane of the meridian. 
Usually, however, there are 
placed in the focus of the eye- 
glass five parallel wires or 
threads, two on each side of 
the central wire, and all at 
equal distances from each oth- 
er, as is represented in the 
following diagram. The time of arriving at each of the wires 
being noted, and all the times added together and divided by 
the number of observations, the result gives the instant of 
crossing the central wire. 

124. The astronomical clock is the constant companion of 
the transit instrument. It is regulated to keep exact pace with 
the stars, and of course with the earth's diurnal rotation ; that 
is, it denotes sidereal time, measuring off one hour for every 15° 
of diurnal motion in a star. The sidereal day begins at the 
moment when the vernal equinox crosses the meridian ; but as 
the culmination of the equinox occurs about 4 minutes later 
from day to day through the year ? the sidereal time may differ 




ASTRONOMICAL INSTRUMENTS. 53 

from the solar time by any quantity whatever. The sidereal 
clock may point to 3h. 20m., in the morning, at noon, or any 
other time of day, because it merely shows that 3h. 20m. have 
elapsed since the equinox was on the meridian. Hence, when 
a star is on the meridian, the sidereal clock shows its right as- 
cension. 

An astronomical clock must have a compensation pendulum, 
and be made as perfect as possible. Its uniformity of move- 
ment can be tested by the transit instrument, and a list of right 
ascensions of stars. It is not so important that it should point 
to Oh. 0m. 0s. when the equinox is on the meridian, or that it 
should not gain or lose compared with the revolution of the 
stars, as that it should move uniformly through the day, and 
from day to day. It is not customary, therefore, to alter the 
clock, after it is once set, but to note from day to day how much 
it is out of the way, and how fast it gains or loses. The first is 
called the error, the last the rate. If these are known, then 
the exact time of an observation can be obtained. 

125. To observe the transit of a star, the eye must discern 
the instant of its bisection by the wire, and the ear attend to 
the beat of the clock, the seconds being counted from the last 
completed minute before the observation began. If the bi- 
section occurs between two beats, as it commonly does, the ob- 
server needs much practice to be able to divide the second 
accurately into tenths, and decide at which of them the transit 
takes place. What is now known as the American method of 
observing transits, and recording them by electro-magnetism, 
gives great facility and accuracy to this most difficult and im- 
portant part of the work. 

The pendulum of the observatory clock is arranged to close 
the circuit of a battery, and break it again at the beginning of 
every beat. The closing of the circuit gives a small lateral 
motion to the registering pen, under which the paper is ad- 
vancing about an inch per second. Thus the seconds are all 
permanently recorded by notches one inch asunder in a straight 

Fig. 15'. 



line, as a, h, c, &c. (Fig. 15'). The mark at the beginning of 
each minute has some peculiarity by which it may be distin- 



54 THE EARTH. 

guished from the rest. The observer has under his hand a key, 
which by a quick touch will also close and break the circuit. 
Whenever a star is on one of the wires of the transit telescope, 
he touches the key, the pen is moved aside and indents the 
line, as at A, and the observation is thus recorded ; and the 
place where this motion commenced between the second-marks 
can afterwards be carefully examined. Thus, without the dis- 
traction of attending to the clock, he can record the transits of 
all the wires ; and if he only notices within what minute the 
work begins, he can read the entire record with accuracy to 
the T V or even the t ^q of a second. Since the general adoption 
of this method, the number of wires in the focus of the eye- 
glass has been increased from 5 to 30 or 40, in order to secure 
a more perfect result. 

126. The vertical circle (n, Fig. 14), usually connected with 
the Transit Instrument, affords the means of measuring arcs on 
the meridian, as meridian altitudes, zenith distances, and decli- 
nations ; but as the circle must necessarily be small, and there- 
fore incapable of measuring very minute angles, the Mural Cir- 
cle is usually employed for measuring arcs of the meridian. 
The Mural Circle is a graduated circle, usually of very large 
size, fixed permanently in the plane of the meridian, and at- 
tached firmly to a perpendicular wall. It is made of large size, 
sometimes 11 feet in diameter, in order that very small angles 
may be measured on its limb ; and it is attached to a massive 
wall of solid masonry in order to insure perfect steadiness, a 
point the more difficult to attain in proportion as the instru- 
ment is heavier. The annexed diagram represents a Mural 
Circle ~B.xed to its wall and ready for observations. It will be 
seen that every expedient is employed to give the instrument 
firmness of parts and steadiness of position. Its radii are com- 
posed of hollow cones, uniting lightness and strength, and its 
telescope revolves on a large horizontal axis, fixed as firmly as 
possible in a solid wall. The graduations are made on the 
outer rim of the instrument, and are read off by six microscopes 
(called reading microscopes), attached to the wall, one of which 
is represented at A, and the places of the five others are marked 
by the letters 'B, C, D, E, F. Six are used, in order that by 
taking the mean of such a number of readings, a higher de- 
gree of accuracy may be insured, than could be obtained by a 



ASTRONOMICAL INSTRUMENTS. 
Fig. 16. 



55 




single reading. In a circle of six feet diameter, like that repre- 
sented in the figure, the divisions may be easily carried to five 
minutes each. The microscope (which is of the variety called 
compound microscope) forms an enlarged image of each of these 
divisions in the focus of the eye-glass. "With it is combined 
the principle of the micrometer. This is effected by placing 
in the focus a delicate wire, which may be moved by means of 
a screw in a direction parallel to the divisions of the limb, and 
which is so adjusted to the screw as to move over the whole 
magnified space of five minutes by five revolutions of the screw. 
Of course one revolution of the screw measures one minute. 
Moreover, if the screw itself is made to carry an index attached 
to its axis and revolving with it over a disk graduated into 
sixty equal parts, then the space measured by moving the in- 
dex over one of these parts will be one second. 

127. We have before shown (Art. 124), the method of find- 
ing the right ascension of a star by means of the Transit In- 



56 



THE EARTH. 



Then its meridian 



strument and the clock. The declination may be obtained by- 
means of the mural circle in several different ways, our object 
being always to find the distance of the star, when on the 
meridian, from the equator (Art. 37). First, the declination 
may be found from the meridian altitude. Let S (Fig. 17) be 
the place of a star when on the meridian, 
altitude will be SH, which will best 
be found by taking its zenith distance 
ZS, of which it is the complement. 
From SH subtract EH, the elevation 
of the equator, which equals the co- 
latitude of the place of observation 
(Art. 44), and the remainder, SE, is the 
declination. Or, if the star is nearer 
the horizon than the equator is, as at 
S', subtract its meridian altitude from 
the co-latitude, for the declination. 
Secondly, the declination may be found from the north polar 
distance, of which it is the complement. Thus from P to E is 
90°. Therefore, PE-PS=90°-PS=SE=the declination. The 
height of the pole P is always known when the latitude of the 
place is known, being equal to the latitude. 




12S. The astronomical instruments already described are 
adapted to taking observations on the meridian only, but we 
sometimes require to know the altitude of a celestial body 
when it is not on the meridian, and its azimuth, or distance 
from the meridian measured on the horizon ; and also the 
angular distance between two points on any part of the celes- 
tial sphere. An instrument especially designed to measure 
altitudes and azimuths, is called an Altitude and Azimuth 
Instrument, whatever may be its particular form. When a 
point is on the horizon, its distance from the meridian, or its 
azimuth, may be taken by the common surveyor's compass, 
the direction of the meridian being determined by the needle ; 
but when the object, as a star, is not on the horizon, its azimuth, 
it must be remembered, is the arc of the horizon from the 
meridian to a vertical circle passing through the star (Art. 27) ; 
at whatever different altitudes, therefore, two stars may be, 
and however the plane which passes through them may be 



ASTRONOMICAL INSTRUMENTS. 57 

inclined to the horizon, still it is their angular distance meas- 
ured on the horizon which determines their difference of 
azimuth. Figure 18 represents an Altitude and Azimuth In- 
strument, several of the usual appendages and subordinate 
contrivances being omitted for the sake of distinctness and 
simplicity. Here abc is the vertical or altitude circle, and EFG 
the horizontal or azimuth circle ; AB is a telescope mounted 

Fig. 18. 




on a horizontal axis and capable of two motions, one in alti- 
tude parallel to the circle abc, and the other in azimuth parallel 
to EFG. Hence it can be easily brought to bear upon any 
object. At m, under the eye-glass of the telescope, is a small 
mirror placed at an angle of 45° with the axis of the telescope, 
by means of which the image of the object is reflected up- 
ward, so as to be conveniently presented to the eye of the ob- 
server. At d is represented a tangent screw, by which a slow 
motion is given to the telescope at c. At h and g are seen two 
spirit-levels at right angles to each other, which show when the 
azimuth circle is truly horizontal. The instrument is supported 
on a tripod, for the sake of greater steadiness, eacli foot being 
furnished with a screw for leveling. 



58 



THE EARTH. 



1 29. The sextant is one of the most useful instruments, both 
to the astronomer and the navigator, and will therefore merit 
particular attention. In figure 19, I and H are two small 
mirrors, and T a small telescope. ID represents a movable 
arm, or radius, which carries an index at D. The radius turns 
on a pivot at I, and the index moves on a graduated arc EF. 

Fig, 19. 




I is called the Index Glass and H the Horizon Glass. The 
under part only of the horizon glass is coated with quicksilver, 
the upper part being left transparent ; so that while one object 
is seen through the upper part by direct vision, another may 
be seen through the lower part by reflection from the two 
mirrors. The instrument is so contrived, that when the index is 
moved up to F, where the zero-point is placed, or the gradu- 
ation begins, the two reflectors I and H are exactly parallel to 
each other. If we now look through the telescope, T, so point- 
ed as to see the star S through the transparent part of the 
horizon glass, we shall see the same star, in the same place, 
reflected from the silvered part ; for the star (or any similar 
object) is at such a distance that the rays of light which strike 
upon the index glass I, are parallel to those which enter the 



ASTRONOMICAL INSTRUMENTS. 59 

eye directly, and will exhibit the object at the same place. 
Now, suppose we wish to measure the angular distance be- 
tween two bodies, as the moon and a star, and let the star be 
at S and the moon at M. The telescope being still directed 
to S, turn the index arm ID from F toward E until the image 
of the moon is brought down to S, its lower limb just touching 
S. By a principle in optics, the angular distance between the 
moon and its image, is twice the angle between the mirrors. 
But the mirror has passed over the graduated arc FD ; there- 
fore double that arc is the angular distance between the star 
and the moon's lower limb. If we then bring the upper limb 
into contact with the star, the sum of both observations, divided 
by 2, will give the angular distance between the star and the 
moon's center. As each degree on the limb EF measures two 
degrees of angular distance, hence the divisions for single 
degrees are in fact only half a degree asunder ; and a sextant, 
or the sixth part of the circle, measures an angular distance of 
120°. The upper and lower points in the disk of the sun or of 
the moon, may be considered as two separate objects, whose 
distance from each other may be taken in a similar manner, 
and thus their apparent diameters at any time be determined. 
We may select our points of observation either in a vertical, or 
in a horizontal direction. 

130. If we make a star, or the limb of the sun or moon, 
one of the objects, and the point in the horizon directly be- 
neath, the other, we thus obtain the altitude of the object. In 
this observation, the horizon is viewed through the transparent 
part of the horizon-glass. At sea, where the horizon is usually 
well defined, the horizon itself may be used for taking altitudes ; 
but on land, inequalities of the earth's surface oblige us to have 
recourse to an artificial horizon. This, in its simple state, is a 
basin of either water or quicksilver. By this means we see 
the image of the sun (or other body) just as far below the 
horizon as it is in reality above it. Hence, if we turn the 
index-glass until the limb of the sun, as reflected from it, is 
brought into contact with the image seen in the artificial 
horizon, we obtain double the altitude.* 



Woodhouse's Ast., p. 774. 



n 



60 THE EARTH. 

The sextant must be held in such a manner that its plane 
shall pass through the plane of the two objects. It must be 
held, therefore, in a vertical plane in taking altitudes, and in a 
horizontal plane in taking the horizontal diameters of the sun 
and moon. Holding the instrument in the true plane of the 
two bodies, whose angular distance is measured, is indeed the 
most difficult part of the operation. 

The peculiar value of the sextant consists in this, that the 
observations taken with it are not affected by any motion in 
the observer ; hence it is the chief instrument used for angular 
measurements at sea. 

131. Examples illustrating the use of the Sextant. 
Ex. 1. Alt. O's lower limb, . . 49° 10' 00 
O's semi-diameter, . . 15 51 

Subtract Refraction, 

Add Parallax, 

True altitude O's center, . 49° 25' 08" 

Ex. 2. With the Artificial Horizon. 
Altitude of ©'s upper limb above the image in the artificial 
horizon, 100° 2' 47". 

True altitude, . . . . . 50° 01' 23".5 
O's semi-diameter, . . . . 00 15 50. 

49° 45' 33".5 
Eefraction, 00 00 48. 

49° 44' 45".5 
Parallax, . 00 00 05. 

True altitude of O's center, . . 49° 44' 50".5 



132. Given the surfs Right Ascension and Declination, to 
find his Longitude and the Obliquity of the Ecliptic. 



49° 


25' 


51" 


00 


00 


49 


49° 


25' 


02" 


00 


00 


06 



* Young's Spherical Trigonometry, p. 136. Vince's Complete System, vol. i. 




ASTRONOMICAL PROBLEMS. 61 

Let POP' (Fig. 20) represent the celestial meridian that 
passes through the first of Cancer and Capricorn (the solstitial 
colure), PP' the axis of the sphere, EQ the equator, E'C the 
ecliptic, and PSP' the declination 
circle (Art. 37) passing through the, Fl s- 20 - 

sun S ; then ARS is a right angle, 
and in the right-angled spherical 
triangle AKS, are given the right 
ascension AK (Art. 37), and the dec- 
lination KS, to find the longitude 
AS and the obliquity SAK. 

As longitude and right ascension 
are measured from A, the first point 
of Aries, in the direction AS of the 
signs, quite round the globe, when, of the four quantities men- 
tioned in the problem, the obliquity and the declination are 
given to find the others, we must know whether the sun is 
north, or whether it is south of the equator, the longitude 
being in the one case AS, and in the other, instead of AS', it 
is 360— AS', when near the same equinox, A. We must also 
know on which side of the tropic the sun is, for the sun in 
passing from one of the tropics to the equinox, passes through 
the same degrees of declination as it had gone through in as- 
cending from the other equinox to the tropic, although its 
longitude and right ascension go on continually increasing. 
From the 21st of March to the 21st of June, while describing 
the firbt quadrant from the vernal equinox, the declination is 
north and increasing; north, but decreasing, in the second 
quadrant, until the 23d of September; south and increasing 
in the third quadrant, until the 21st of December ; and finally, 
in the fourth quadrant, south but decreasing until the 21st of 
March. 

Ex. 1. On the 17th of May, the sun's Right Ascension was 
53° 38', and his Declination 19° 15' 57" : required his Longitude 
and the Obliquity of the Ecliptic. 

Applying Napie^s Mule* to the right-angled triangle APS, 
we have 



* The student is supposed to be acquainted with Spherical Trigonometry; but 
to refresh his memory, we may insert a remark or two. 



62 



THE EARTH. 



1. Rad cos AS = cos AR cos RS. 

2. Rad sin AE=tan RS cot A.-, cot A= _ rad * m AR 

tan RS 

Hence the computation for AS and A is as follows : 



For the Longitude AS. 

cos AR 53° 38' 00" 9.7730185 
cosRS 19 15 57 9.9749710 



cos AS 55 57 43 9.7479895 



For the Obliquity A. 

sinAR 9.9059247 

tan RS, ar. com. 0.4565209 



cot A 23° 27' 50i' 



10.3624456 



Fie. 21. 




It will be recollected that in Napier's rule for the solution of a right-angled 
spherical triangle, by means of the Five Circular Parts, we proceed as follows : 

In a right-angled spherical triangle we are to recognize but five parts, viz., the 
three sides and the two oblique angles. If we take any one of these as a middle 
part, the two which lie next to it on each side will be adjacent parts. Thus (in 
Fig. 21), taking A for a middle part, b and c will be the adjacent parts ; if we 
take c for the middle part, A and B will 
be the adjacent parts ; if we take B for 
the middle part, c and a will be the ad- 
jacent parts ; but if we take a for the 
middle part, then as the angle C is not 
considered as one of the circular parts, B 
and b are the adjacent parts ; and, lastly, 
if b is the middle part, then the adjacent 
parts are A and a. The two parts immediately beyond the adjacent parts on 
each side, still disregarding the right angle, are called the opposite parts ; thus, 
if A is the middle part, the opposite parts are a and B. Napier's rule is as 
follows : 

Radius into the sine of the middle part, equals the product of the tangents of the adjacent 
parts, or of the cosines of Ike opposite parts. 

(The corresponding vowels are marked to aid the memory ) In the use of this 
rule, it must be understood that the complements of the angles and the hypote- 
nuse are used, instead of those parts themselves. Thus, if A is middle part, we 
say rad X cos A, not rad X sin A, and sc of B ; or, if A B is adjacent part, we use 
cot AB, not tan AB ; if opposite, sin AB, not cos AB, &c 

Examples. 1. In the right-angled triangle ABC, are given the two perpen- 
dicular sides, viz., a=48° 24' 10'', 6=59° 88' 27", to find the hypotenuse c. The 
hypotenuse being made the middle part, the other sides become the opposite 
parts, being separated from the middle part by the angles A and B. Hence, 



cos a cos b 

Rad cos c=cos a cos b .\ cos c= -% =70 28 40 " . 

rad 

2. In the spherical triangle, right-angled at C, are given two perpendicular 
sides, viz., a— 116° 30' 43", 6=29° 41' 32", to find the angle A. 

Here the required angle is adjacent to one of the given parts, viz., b, which make 
the middle part. Then, 



RadXsin 6=cot A tan a 



radXsm b 

cot A=— =/ 6 

tan a 



13". 



ASTRONOMICAL PROBLEMS. 63 

Ex. 2. On the 31st of March, 1816, the sun's Declination was 
observed at Greenwich to be 4° 13' 31-J-": required his Right 
Ascension, the obliquity of the ecliptic being 23° 27' 51". 

Ans. 9° 47' 59". ' 
Ex. 3. What was the sun's Longitude on the 28th of Novem- 
ber, 1810, when his Declination was 21° 16' 4", and his Right 
Ascension, in time, 16h. 14m. 58.4s. ? 

Ans. 245° 39' 10". 

Ex. 4. The sun's Longitude being 8 s 7° 40' 56", and the 
Obliquity 23° 27' 42^", what was the Right Ascension in 
time? Ans. 16h. 23m. 34s. 

133. Given the sun's Declination to find the time of his 
Rising and Setting at any place whose latitude is known. 

Let PEP' (Fig. 22) represent the meridian of the place, Z 
being the zenith, and HO the horizon ; and let LL' be the ap- 
parent path of the sun on the pro- 
posed day, cutting the horizon in S. 
Then the arc EZ will be the latitude 
of the place, and consequently EH, 
or its equal QO, will be the co-lati- 
tude, and this measures the angle 
OAQ ; also RS will be the sun's dec- 
lination, and AR expressed in time 
will be the time of rising before 6 
o'clock. For it is evident that it 
will be sunrise when the sun arrives at the horizon at S ; but 
PP' being an hour circle whose plane is perpendicular to the 
meridian (and of course projected into a straight line on the 
plane of projection), the time the sun is passing from S to S' 
taken from the time of describing S'L, which is six hours, must 
be the time from midnight to sunrise. But the time of de- 
scribing SS' is measured on the corresponding arc of the equi- 
noctial AR. 

In the right-angled triangle ARS, we have the declination 
RS, and the angle A to find AR. Therefore, 

Radxsin AR=cot Ax tan RS. 

Ex. 1. Required the time of sunrise at latitude 52° 13' N. 
when the sun's declination is 23° 28'. 




64 



THE EARTH. 



Rad 

Cot A or tan 52° 13' 
Tan KS= 23° 28' 
Sin 34° 03' 21 J" 

4* 



10. 

10.1105786 
9.6376106 

9.7481892 



2h. 16' 13" 25' 
6 



3h. 43' 46" 35'"= the time after midnight, and of 

course the time of rising. 

Ex. 2, Kequired the time of sunrise at latitude 57° 2' 54" N. 

when the sun's declination is 23° 28' N. 

Ans. 3h. 11m. 49s. 

Ex. 3. How long is the sun above the horizon in latitude 58° 
12' K when his declination is 18° 40' S. ? 

Ans. 7h. 35m. 52s. 



134. Given the Latitude of the place, and the Declination 
of a heavenly body, to determine its Altitude and Azimuth 
when on the six o'* clock hour circle. 

Let HZO (Fig. 23) fee the meridian of the place/Z the zenith, 
HO the horizon, S the place of the object on the 6 o'clock hour 



Fig. 23. 



circle PSP', which of course 
cuts the equator in the east and 
west points, and ZSB the verti- 
cal circle passing through the 
body. Then in the right-angled 
triangle SB A, the given quan- 
tities are AS, which is the dec- 
lination, and the arc OP or 
angle SAB, the latitude of the 
place, to find the altitude BS, 
and the azimuth BO, or the 
amplitude AB, which is its 
complement. 

Ex. 1. "What were the altitude and azimuth of Arcturus, 
when upon the six o'clock hour circle of Greenwich, lat. 51° 




° Degrees are converted into hours by multiplying by 4 and dividing by 60. 



ASTRONOMICAL PROBLEMS. 



65 



28' 40 " N. on the first of April, 1822 ; its declination being 

20° 6' 50" 1ST. ? 



For the Altitude. 

Bad sin BS=sin AS sin A 
Rad . . 10. 
Sin 20° 06' 50" 9.5364162 
Sin 51 28 40 9.8934103 

Sin 15 36 27 9.4298265 



For the Azimuth. 

Rad cos A=cot BO cot AS 
Cot 20° 06' 50" 10.4362545 
Cos 51 28 40 9.7943612 

Rad . . 10. 



Cot 77° 09' 04' 



9.3581067 



Ex. 2. At latitude 62° 12' K the altitude of the sun at 6 
o'clock in the morning was found to be 18° 20' 23": required 
his declination and azimuth. 

Ans. Dec. 20° 50' 12" 1ST. Az. 79° 56' 4". 

135. The Latitudes and Longitudes of two celestial objects 
being given, to find their Distance ajpavt. 

Let P (Fig. 24) represent the pole of the ecliptic, and PS, 
PS', two arcs of celestial latitude (Art. 37) drawn to the two 



Fig. 24. 




objects SS'; then will these arcs rep- 
resent the co-latitudes, the angle P 
will be the difference of longitude, 
and the arc SS' will be the distance 
sought. Here we have the two sides 
and the included angle given to find 
the third side. By Napier's Rules 
for the solution of oblique-angled spherical triangles (see 
Spherical Trigonometry), the sum and difference of the two 
angles opposite the given sides may be found, and thence the 
angles themselves. The required side may then be found by 
the theorem, that the sines of the sides are as the sines of their 
opposite angles.* The computation is omitted here on account 
of its great length. If P be the pole of the equator instead of 
the ecliptic, then PS and PS' will represent arcs of co-declina- 
tion, and the angle P will denote difference of right ascension. 
From these data, also, we may therefore derive the distance 
between any two stars. Or, finally, if P be the pole of the 
horizon, the angle at P will denote difference of azimuth, and 

s More concise formulas for the solution of this case may be found in Young's 
Trigonometry, p. 99 ; Francceur's Uranography, Art. 830 ; Dr. Bowditch's 
Practical Navigator, p. 436. 

5 



Ob THE EARTH. 

the sides PS, PS', zenith distances, from which the side SS' 
may likewise be determined. 

FIGURE AND DENSITY OF THE EARTH. 

136. "We have already shown (Art. 8) that the figure of the 
earth is nearly globular; but since the semi-diameter of the 
earth is taken as the base line in determining the parallax of 
the heavenly bodies, and lies, therefore, at the foundation of 
all astronomical measurements, it is very important that it 
should be ascertained with the greatest possible exactness. 
Having now learned the use of astronomical instruments, and 
the method of measuring arcs on the celestial sphere, we are 
prepared to understand the methods employed to determine 
the exact figure of the earth. This element is indeed ascer- 
tained in five different wavs, each of which is independent of 
all the rest, namely, by investigating the effects of the centrifu- 
gal force arising from the revolution of the earth on its axis — 
by measuring arcs of the meridian — by experiments with the 
pendulum — by the unequal action of the earth on the moon, 
arising from the redundance of matter about the equatorial 
regions — and by the precession of the equinoxes. "We will 
briefly consider each of these methods. 

137. First, the known effects of the centrifugal force would 
give to the earth a spheroidal figure, elevated in the equatorial, 
and flattened in the polar regions. 

Had the earth been originally constituted (as geologists sup- 
pose) of yielding materials, either fluid or semi-fluid, so that 
its particles could obey their mutual attraction, while the body 
remained at rest it would spontaneously assume the figure of a 
perfect sphere ; as soon, however, as it began to revolve on its 
axis, the greater velocity of the equatorial regions would give 
to them a greater centrifugal force, and cause the body to 
swell out into the form of an oblate spheroid.* Even had the 
solid part of the earth consisted of unyielding materials and 
been created a perfect sphere, still the waters that covered it 
would have receded from the polar and have been accumulated 



* See a good explanation of this subject in the Edinburgh Encyclopaedia, ii., 665. 



FIGURE OF THE EARTH. 



67 



in the equatorial regions, leaving bare extensive regions on the 
one side, and ascending to a mountainous elevation on the other. 
On estimating from the known dimensions of the earth and 
the velocity of its rotation, the amount of the centrifugal force 
in different latitudes, and the figure of equilibrium which would 
result, Newton inferred that the earth must have the form of 
an oblate spheroid before the fact had been established by ob- 
servation ; and he assigned nearly the true ratio of the polar 
and equatorial diameters. 

138. Secondly, the spheroidal figure of the earth is proved 
by actually measuring the length of a degree on the meridian in 
different latitudes. 

Were the earth a perfect sphere, the section of it made by a 
plane passing through its center in any direction would be a 
perfect circle, whose curvature would be equal in all parts ; 
but if we find by actual observation that the curvature of the 
section is not uniform, we infer a corresponding departure in 
the earth from the figure of a perfect sphere. This task of 
measuring portions of the meridian has been executed in dif- 
ferent countries by means of a system of triangles with aston- 
ishing accuracy.* The result is, that the length of a degree 
increases as we proceed from the equator toward the pole, as 
may be seen from the following table : 



Places of observation. 


Latitude. 


Length of a degree 
in miles. 


Peru 


00° 00 f 00" 
39 12 00 
43 01 00 
46 12 00 
51 29 54| 
QQ 20 10 


68/732 
68.896 
68.998 
69.054 
69.146 
69.292 


Pennsylvania . . . 

Italy 

France 


England 

Sweden 



Combining the results of various measurements, the dimen- 
sions of the terrestrial spheroid are found to be as follows : f 

Equatorial diameter, 7925.308 

Polar diameter, 7898.952 

Mean diameter, 7912.1 30 



* See Day's Trigonometry. 



f Bessel. 



68 THE EARTH. 

The difference between the greatest and least, is 26.356 = 3^ 
of the greatest. This fraction (3-^7) is denominated the ellip- 
ticity of the earth, being the excess of the transverse over the 
conjugate axis, on the supposition that the section of the earth 
coinciding with the meridian is an ellipse ; and that such is 
the case, is proved by the fact that calculations on this hypoth- 
esis, of the lengths of arcs of the meridian in different latitudes, 
agree nearly with the lengths obtained by actual measurement. 

139. Thirdly, the figure of the earth is shown to he spheroidal 
by observations with the pendulum. 

The use of the pendulum in determining the figure of the 
earth, is founded upon the principle that the number of vibra- 
tions performed by the same pendulum, when acted on by dif- 
ferent forces, varies as the square root of the forces* Hence, 
by carrying a pendulum to different parts of the earth, and 
counting the number of vibrations it performs in a given time, 
we obtain the relative forces of gravity at those places ; and 
this leads to a knowledge of the relative distance of each place 
from the center of the earth, and finally, to the ratio between 
the equatorial and the polar diameters, 

1 40. Fourthly, that the earth is of a spheroidal figure is 
mf erred from the motions of the moon. 

These are found to be affected by the excess of matter about 
the equatorial regions, producing certain irregularities in the 
lunar motions, the amount of which becomes a measure of the 
excess itself, and hence affords the means of determining the 
earth's ellipticity. This calculation has been made by the most 
profound mathematicians, and the figure deduced from this 
source corresponds very nearly to that derived from the several 
other independent methods. 

141. Fifthly, the spheroidal farm of the earth accounts for 
the precession of the equinoxes. 

It will be shown in Chap. IT., Part II., that the slow back- 
ward motion of the equinoctial points on the ecliptic is due to 
the attraction of the sun and moon upon the belt of matter on 

* Mechanics, Art. 161. 



.DENSITY OF THE EARTH. 



69 



the equator, combined with the inertia of the earth in its rota- 
tion on its axis. 

We thus have the shape of the earth established upon the 
most satisfactory evidence, and are furnished with a starting 
point from which to determine various measurements among 
the heavenly bodies. 



&-& 



Fig. 25. 
1 



141'. The density of the earth compared with water, that 
is, its specific gravity, is 5^.* The density was first estimated 
by Dr. Hutton, from observations made by Dr. Maskelyne, 
Astronomer Royal, on Sehehallien, a mountain of Scotland, in 
the year 177-1. Thus, let M 
(Fig. 25) represent the moun- 
tain, D, B, two stations on op- 
posite sides of the mountain, 
and I a star; and let IE and 
IG be the zenith distances as 
determined by the differences 
of latitudes of the two stations. 
But the apparent zenith dis- 
tances as determined by the 
plumb-line are IE' and IG'. 
The deviation toward the moun- 
tain on each side exceeded 7".f 
The attraction of the mountain 
being observed on both sides 
of it, and its mass being computed from the number of sec- 
tions taken -in all directions, these data, when compared with 
the known attraction and magnitude of the earth, led to a 
knowledge of its mean density. According to Dr. Hutton, 
this is to that of water as 9 to 2 ; but later and more accurate 
estimates have made the specific gravity of the earth as stated 
above. But this density is nearly double the average density 
of the materials that compose the exterior crust of the earth, 
showing a great increase of density toward the center. 

The density of the earth is an important element, as we shall 
find that it helps us to a knowledge of the density of each of 
the other members of the solar system. 





~ Baily, Ast. Tables, p. 21. 



f Eobison's Phys. Ast. 



PART II.— OF THE SOLAR SYSTEM 



142. Haying- considered the Earth, in its astronomical re- 
lations, and the Doctrine of the Sphere, we proceed now to a 
survey of the Solar System, and shall treat successively of the 
Sun, Moon, Planets, and Comets. 



CHAPTEE I. 

OF THE SUN" SOLAK SPOTS — ZODIACAL LIGHT. 

143. The figure which the sun presents to us is that of a 
perfect circle, whereas most of the planets exhibit a disk more 
or less elliptical, indicating that the true shape of the body is 
an oblate spheroid. So great, however, is the distance of the 
sun, that a line 400 miles long would subtend an angle of only 
1" at the eye, and would, therefore, be the least space that 
could be measured. Hence, were the difference between two 
conjugate diameters of the sun any quantity less than this, we 
could not determine by actual measurement that it existed at 
all. Still we learn from theoretical considerations, founded 
upon the known effects of centrifugal force, arising from the 
sun's revolution on his axis, that his figure is not a perfect 
sphere, but is slightly spheroidal.* 

144. The distance of the sun from, the earth is nearly 
95,000,000 miles. For, its horizontal parallax being 8".6 
(Art. 86), and the semi-diameter of the earth 3956 miles, 

Sin 8".6 : 3956 : : Rad : 95,000,000 nearly. 
In order to form some faint conception at least of this vast 
distance, let us reflect that a railway car, moving at the rate of 
20 miles per hour, would require more than 500 years to reach 
the sun. 

s See Me'canique Celeste, iii., p. 165. Delambre, t. i., p. 483. 



SOLAR SPOTS. 



71 



145. The apparent diameter of the sun may be found either 
by the Sextant (Art. 129), by an instrument called the Heli- 
ometer, specially designed for measuring its angular breadth, 
or by the time it occupies in crossing the meridian. 

The last is the most accurate. If the sun, when on the 
equator, March 21 or September 22, is found to cross the me- 
ridian in 2m. 10s., sidereal time* then 24h. : 2m. 10s. : : 360° : 32' 
30"= the angular breadth of the sun. But if the sun ha's a dec- 
lination north or south, the degrees of the diurnal circle on 
which he moves are shorter than those on the equator, in the 
ratio of the cosine of declination to radius ; and therefore the 
time of crossing is lengthened; hence the calculated breadth 
must be diminished in the same ratio. . Having the distance 
and angular diameter, we can easily find its ■ 
linear diameter. Let E (Fig. 26) be the 
earth, S the sun, ES a line drawn to the 
center of the disk, and EC a line drawn' 
touching the disk at C. Join SG ; then 

Kad : ES : : sin 16' 1".5 (the sun's mean 
apparent semidiameter) : SC =442,840 miles. 

2x442840 • 

= 112 nearly ; that is, 



Fig. 26. 



And 



7912 



it 




would require one hundred and twelve bodies 
like the earth, if laid side by side, to reach 
across the diameter of the sun. Since spheres 
are to each other as the cubes of their diam- 
eters, I s : 112 3 : : 1 : 1,400,000 nearly ; that 
is, the sun is- about 1,400,000 times as large as the earth. 

146. In density ', the sun is only one-fourth that of the 
earth, being but a little heavier than water (Art. 141 ') ; and 
since the quantity of matter, or mass of a body, is proportion- 
ed to its magnitude and density, hence 1,400,000x^=350,000; 
that is, the quantity of matter in the sun is three hundred and 
fifty thousand (or, more accurately, 354,936) times as great as 
in the earth. Now the weight of bodies (which is a measure 
of the force of gravity) varies directly as the quantity of mat- 
ter, and inversely as the square of the distance. A body, 
therefore, would weigh 350,000 times as much on the surface 
of the sun as on the earth, if the distance of the center of force 



72 



THE SUN. 



were the same in both cases; but since the attraction of a 
sphere is the same as though all the matter were collected in 
the center, consequently, the weight of a body, so far as it de- 
pends on its distance from the center of force, would be the 
square of 112 times less at the sun than at the earth. Or, put- 
ting W for the weight at the .earth, and W for the weight at 
the sun, then 

Hence a body would weigh nearly 28 times as much at the 
sun as at the earth. A man weighing 200 lbs. would, if trans- 
ported to the surface of the sun, weigh 5,580 lbs., or nearly 2J- 
tons. To lift one's limbs would, in such a case, be beyond the 
ordinary power of the muscles. At the surface of the earth a 
body falls through 16 T ^- feet in a second ; and since the spaces 
are as the velocities, the times being equal, and the velocities 
as the forces, therefore a body would fall at the sun in one 
second through 16 T 1 2x27 T 9 o =448.7 feet. 

SOLAR SPOTS. 

* 

147. The surface of the sun, when viewed with a tele- 
scope, often shows dark spots, which vary much, at different 
times, in number, figure, and extent. One hundred or more, 
assembled in several distinct groups, are sometimes visible at 
once on the solar disk. The solar spots are commonly very 
small, but occasionally a spot of enormous size is seen occupy- 
ing an extent of 50,000 miles or more in diameter. They are 
sometimes even visible to the naked eye, when the sun is 
viewed through colored glass, or when near the horizon, it is 
seen through light clouds or vapors. "When it is recollected 
that 1" of the solar disk implies an extent of 400 miles (Art. 
143), it is evident that a space large enough to be seen by the 
naked eye, must cover a very large extent. 

Fig. 27 exhibits the appearance of solar spots, though much 
too large compared with the disk. Whenever a spot is seen near 
the edge of the disk, it appears foreshortened by perspective, 
as in the figure. A solar spot usually consists of two parts, 
the nucleus and the umhra. The nucleus is black, of a very 
irregular shape, and is subject to great and sudden changes 



SOLAR SPOTS. 



73 




both in form and size. A Fi S- 27- 

spot sometimes divides into 

many smaller ones, and again 

a group may be united into a 

single spot. The umbra is a 

wide margin of lighter shade, 

and is commonly of greater 

extent than the nucleus. The 

spots are usually confined to 

a zone extending across the 

central regions of the sun, not 

exceeding 60° in breadth. When the spots are observed from 

day to day, they are seen to move across the disk of the sun, 

occupying about two weeks in passing from one limb to the 

other. After an absence of about the same period, the spot 

returns, having taken 27d. 7h. 37m. in the entire revolution. 

148. Besides the fact of foreshortening already mentioned, 
there is another proof that the spots are at the surface of the 
sun. Were they bodies at a distance 
from it, the time during which they 
would be seen on the solar disk would 
be less than that occupied in the re- 
mainder of the revolution. Thus, let S 
(Fig. 28) be the sun, E the earth, and 
dbc the path of the body revolving 
about the sun. Unless the spot were 
nearly or quite in contact with the 
body of the sun, being projected upon 
his disk only while passing from o to 
c, and being invisible while describing 
the arc cab, it would of course be out 
of sight longer than in sight, whereas 
the two periods are found to be equal. 
Moreover, the lines which all the solar 
spots describe on the disk of the sun 
are found to be parallel to each other, 
like the circles of diurnal revolution 
around the earth ; and hence it is in- 
ferred that they arise from a similar cause, namely, the rcrohi- 



Fig. 28. 





74: THE SUN. 

tion of the sun on his axis, a fact which is thus made known 
to us. 

But although the spots occupy about 27i days in passing 
from one limb of the sun around to the same limb again, yet 
this is not the period of the sun's revolution on his axis, but 
exceeds it by nearly two days. For, let AA'B (Fig. 29) repre- 
sent the sun, and EE'M the orbit of the earth. When the 
earth is at E, the visible disk of the 
sun will be AA'B ; and if the earth 
remained stationary at E, the time oc- 
cupied by a spot after leaving A until 
it returned to A, would be just equal 
to the time of the sun's revolution on 
his axis. But during the 27^ days in 
which the spot has been performing 
its apparent revolution, the earth has 
been advancing in her orbit from E to 
E', where the visible disk of the sun is 
A'B'. Consequently, before the spot 

can appear again on the limb from which it set out, it must 
describe so much more than an entire revolution as equals the 
arc AA', which equals the arc EE'. Hence, 

365d. 5h. 4Sm. + 27d. 7h. 37m. : 365d. 5h. 48m. :: 27d. Yh. 
37m. : 25d. 9h. 59m.=the time of the sun's revolution on 
his axis. 

1 49. If the path which the spots appear to describe by the 
revolution of the sun on his axis left each a visible trace on his 
surface, they would form, like the circles of diurnal revolution 
on the earth, so many parallel rings, of which that which 
passed through the center would constitute the solar equator, 
while those on each side of this great circle would be small 
circles, corresponding to parallels of latitude on the earth. Let 
us conceive of an artificial sphere to represent the sun, having 
such rings plainly marked on its surface. Let this sphere be 
placed at some distance from the eye, with its axis perpendicu- 
lar to the axis of vision, in which case the equator would coin- 
cide with the line of vision, and its edge be presented to the 
eye. It would therefore be projected into a straight line. The 
same would be the case with all the smaller rings, the distance 



SOLAR SPOTS. 75 

being supposed such that the rays of light come from them all 
to the eye nearly parallel. Now let the axis, instead of being 
perpendicular to the line of vision, be inclined to that line, 
then all the rings being seen obliquely, would be projected 
into ellipses. If, however, while the sphere remained in a 
fixed position, the eye were carried around it (being always in 
the same plane) twice during the circuit, it would be in the 
plane of the equator, and project this and all the smaller 
circles into straight lines ; and twice, at points 90° distant from 
the foregoing positions, the eye would be at a distance from 
the planes of the rings equal to the inclination of the equator 
of the sphere to the line of vision. Here it would project the 
rings into wider ellipses than at other points ; and the ellipses 
would become more and more eccentric as the eye departed 
from either of these points, until they vanished again into 
straight lines. 

150. It is in a similar manner that the eye views the paths 
described by the spots on the sun. If the sun revolved on an 
axis perpendicular to the plane of the earth's orbit, the eye be- 
ing situated in the plane of revolution, and at such a distance 
from the sun that the light comes to the eye from all parts of 
the solar disk nearly parallel, the paths described by the spots 
would be projected into straight lines, and each would describe 
a straight line across the solar disk, parallel to the plane of 
revolution. But the axis of the sun is inclined to the ecliptic 
about Yi° from a perpendicular, so that usually all the circles 
described by the spots are projected into ellipses. The breadth 
of these, however, will vary as the eye, in the annual revolu- 
tion, is carried around the sun, and when the eye comes into 
the plane of the rings, as it does twice a year, they are pro- 
jected into straight lines, and for a short time a spot seems 
moving in a straight line inclined to the plane of the ecliptic 
7i°. The two points where the sun's equator cuts the ecliptic 
are called the sun's nodes. The longitudes of the nodes are 
80° V and 260° 7', and the earth passes through them about 
the 12th of December, and the 11th of June. It is at these 
times that the spots appear to describe straight lines. We 
have mentioned the various changes in the apparent paths of 
the solar spots, which arise from the inclination of the sun's 



76 



THE SUN. 



axis to the plane of the ecliptic ; but it was in fact by first ob- 
serving these changes, and proceeding in the reverse order 
from that which we have pursued, that astronomers ascer- 
tained that the sun revolves on his axis, and that this axis is 
inclined to the ecliptic 82f°. 

151. Besides the spots already described, there are faint 
inequalities of light over the general surface, in delicate lines 
and freckles, which are also perpetually changing. These are 
called faculcB. 

The theory, which is generally received, regards the visible 
surface of the sun as an incandescent, gaseous substance, 
always in violent commotion, and sometimes rent here and 
there by a broad opening, which reveals a lower stratum of 
less illumination. This is the umbra of a spot ; and a smaller 
rupture in that shows a still lower stratum, or else the solid 
body of the sun, as the 'nucleus of the same. These apertures 
in the luminous strata may be caused, as some think, by vol- 
canic action below, or according to others, by storms or tor- 
nadoes in the solar atmosphere. 



ZODIACAL LIGHT. 



152. The Zodiacal Light is 
a faint light resembling the tail 
of a comet, and is seen at cer- 
tain seasons of the year following 
the course of the sun after even- 
ing twilight, or preceding his 
approach in the morning sky. 
Figure 30 represents its appear- 
ance as seen in the evening, in 
March, 1836. The following are 
the leading facts respecting it. 

1. Its form is that of a lumi- 
nous triangle, having its base 
toward the sun. It reaches to 
an immense distance from the 
sun, sometimes even beyond the 



Fig. 30. 




ZODIACAL LIGHT. 77 

orbit of the earth. It is brighter in the parts nearer the sun 
than in those that are more remote, and terminates in an ob- 
tuse apex, its light fading away by insensible gradations, until 
it becomes too feeble for distinct vision. Hence its limits are, at 
the same time, fixed at different distances from the sun by dif- 
ferent observers, according to their respective powers of vision. 

2. Its aspects vary very much with the different seasons of the 
year. About the first of October, in our climate (Lat. 41° 
18'), it becomes visible before the dawn of day, rising along 
north of the ecliptic, and terminating above the nebula of 
Cancer. About the middle of November, its vertex is in the 
constellation Leo. At this time no traces of it are seen in the 
west after sunset, but about the first of December it becomes 
faintly visible in the west, crossing the Milky Way near the 
horizon, and reaching from the sun to the head of Capri- 
cornus, forming, as its brightness increases, a counterpart to 
the Milky Way, between which on the right, and the Zodiacal 
Light on the left, lies a triangular space embracing the Dol- 
phin. Through the month of December, the Zodiacal Light is 
seen on both sides of the sun, namely, before the morning 
and after the evening twilight, sometimes extending 50° west- 
ward, and 70° eastward of the sun at the same time. After it 
begins to appear in the western sky, it increases rapidly 
from night to night, both in length and brightness, and with- 
draws itself from the morning sky, where it is scarcely seen 
after the month of December, until the next October. 

3. The Zodiacal Light moves through the heavens in the 
order of the signs. It moves with unequal velocity, being 
sometimes stationary and sometimes retrograde, while at other 
times it advances much faster than the sun. In February and 
March, it is very conspicuous in the west, reaching to the 
Pleiades and beyond ; but in April it becomes more faint, and 
nearly or quite disappears during the month of May. It is 
scarcely seen in this latitude during the summer months. 

4. It is remarkably conspicuous at ■certain periods of a few 
years, and then for a long interval almost disappears. 

5. The Zodiacal Light toas formerly held to he the atmos- 
phere of the sun.* But La Place has shown that the solar 

* Mairan, Memoirs French Academy, for 1783. 



78 THE STTN. 

atmosphere could never reach so far from the sun as this light 
is seen to extend.* It has been supposed by others to be a 
nebulous body revolving around the sun. From recent obser- 
vations, made with care in various parts of tropical America, 
there appears to be strong evidence that the Zodiacal Light is 
a belt which entirely surrounds the earth.f 



CHAPTEK II. 

OF THE APPAEENT ANNUAL MOTION OF THE SUN SEASONS— FIG- 



153. The revolution of the earth around the sun once a 
year, produces an apparent motion of the sun around the earth 
in the same period. When bodies are at such a distance from 
each other as the earth and the sun, a spectator on either 
would project the other body upon the concave sphere of the 
heavens, always seeing it on the opposite side of a great circle, 
180° from himself. Thus, when the earth arrives at Libra 
(Fig. 11), we see the sun in the opposite sign Aries. When 
the earth moves from Libra to Scorpio, as we are unconscious 
of our own motion, the sun it is that appears to move from 
Aries to Taurus, being always seen in the heavens, where a 
line drawn from the eye of the spectator through the body 
meets the concave sphere of the heavens. Hence the line of 
projection carries the sun forward on one side of the ecliptic, 
at the same rate as the earth moves on the opposite side ; and 
therefore, although we are unconscious of our own motion, we 
can read it from day to day in the motions of the sun. If we 
could see the stars at the same time with the sun, we could 
actually observe from day to day the sun's progress through 
them, as we observe the progress of the moon at night ; only 
the sun's rate of motion would be nearly fourteen times slower 
than that of the moon. Although we do not see the stars 

* Mec. Celeste, iii., 525. 

f See a paper by Rev. Geo. Jones, U. S. Navj'. Proc. Amer. Assoc, 1859, p. 
172. 



ANNUAL ' MOTION. i \) 

when the sun is present, yet after the sun is set, we can ob- 
serve that it makes daily progress eastward, as is apparent 
from the constellations of the Zodiac occupying, successively, 
the western sky after sunset, proving that either all the stars 
have a common motion westward independent of their diurnal 
motion, or that the sun has a motion past them, from west to 
east. TTe shall see hereafter abundant evidence to prove that 
this change in the relative position of the sun and stars, is 
owing to a parallactic change in the place of the sun, and not 
to any change in the stars. 

« 

154. Although the apparent revolution of the sun is in a 
direction opposite to the real motion of the earth, as regards 
absolute space, yet both are nevertheless from west to east, 
since these terms do not refer to any directions in absolute 
space, but to the order in which certain constellations (the con- 
stellations of the Zodiac) succeed one another. The earth itself, 
on opposite sides of its orbit, does, in fact, move toward di- 
rectly opposite points of space ; but it is all the while pursuing 
its course in the order of the signs. In the same manner, 
although the earth turns on its axis from west to east, yet any 
place on the surface of the earth is moving in a direction in 
space exactly opposite to its direction twelve hours before. If 
the sun left a visible trace on the face of the sky, the ecliptic 
would, of course, be distinctly marked on the celestial sphere 
as it js, on an artificial globe ; and were the equator delineated 
in a similar manner (by any method like that supposed in Art. 
46), we should then see at a glance the relative position of 
these two circles ; the points where they intersect one another 
constituting the equinoxes, the points where they are at the 
greatest distance asunder, or the solstices, and various other 
particulars, which, for want of such visible traces, we are now 
obliged to search for by indirect and circuitous methods. It 
will even aid the learner to have constantly before his mental 
vision, an imaginary delineation of these two important circles 
on the face of the sky. 

155. The 7nethod of ascertaining the nature and 'position of 
the earth's orbit is by observations on the sun's Declination and 
Right Ascension. 



80 THE SUN. 

The exact declination of the sun at any time is determined 
from his meridian altitude or zenith distance, the latitude of 
the place of observation being known (Art. 37). The instant 
the center of the sun is on the meridian (which instant is given 
by the transit instrument), we take the distance of his upper 
and that of his lower limb from the zenith : half the sum of the 
two observations corrected for refraction, gives the zenith dis- 
tance of the center. This result is diminished for parallax 
(Art. 84), and we obtain the zenith distance as it would be if 
seen from the center of the earth. The zenith distance being 
known, the declination is readily found by subtracting that 
distance from the latitude. By thus taking the sun's declina- 
tion for every day of the year at noon, and comparing the 
results, we learn its motion to and from the equator. 

156. To obtain the motion in right ascension, we observe, 
with a transit instrument, the instant when the center of the 
sun is on the meridian. Our sidereal clock gives us the right 
ascension in time (Art. 124), which we may easily, if we 
choose, convert into degrees and minutes, although it is more 
common to express right ascension by hours, minutes, and 
seconds. The differences of right ascension from day to day 
throughout the year, give us the sun's annual motion parallel 
to the equator. From the daily records of these two motions, 
at right angles to each other, arranged in a table,* it is easy to 
trace out the path of the sun on the artificial globe ; or to cal- 
culate it with the greatest precision by means of spherical tri- 
angles, since the declination and right ascension constitute two 
sides of a right-angled spherical triangle, the corresponding arc 
of the ecliptic, that is, the longitude, being the third side (Art. 
132). By inspecting a table of observations, we shall find that 
the declination attains its greatest value on the 22d of Decem- 
ber, when it is 23° 27' 51" south ; that from this period it di- 
minishes daily and becomes nothing on the 21st of March; 
that it then increases toward the north, and reaches a similar 
maximum at the northern tropic about the 22d of June ; and, 
finally, that it returns again to the southern tropic by grada- 



* Such a table may be found in Biot's Astronomy, in Delambre, and in most 
collections of Astronomical Tables. 



ANNUAL MOTION. 81 

tions similar to those which marked its northward progress. 
A table of observations, also, would show us that the daily 
differences of declination are very unequal ; that, for several 
days, when the sun is near either tropic, its declination scarcely 
varies at all ; while near the equator, the variations from day 
to day are very rapid — a fact which is easily understood, when 
we reflect, that at the solstices the equator and the ecliptic are 
parallel to each other,* both being at right angles to the 
meridian ; while at the equinoxes, the ecliptic departs most 
rapidly from the direction of the equator. 

On examining, in like manner, a table of observations of 
the right ascension, we And that the daily differences of right 
ascension are likewise unequal ; that the mean of them all is 
3 m 56 s , or 236 s , but that they have varied between 215 s and 
266\ On examining, moreover, the right ascension at each of 
the equinoxes, we find that the two records differ by 180°; 
which proves that the path of the sun is a great circle, since 
no other would bisect the equinoctial as this does. 

157. The obliquity of the ecliptic is equal to the surfs great- 
est declination. For, by article 22, the inclination of any two 
great circles is equal to their greatest distance asunder, as 
measured on the sphere. The obliquity of the ecliptic may be 
determined from the sun's meridian altitude, or zenith distance, 
on the day of the solstice. The exact instant of the solstice, 
however, is not likely to occur when the sun is on the merid- 
ian, but may happen at some other meridian \ still, the changes 
of declination near the solstice are so exceedingly small that 
but a slight error can result from this source. The obliquity 
may also be found, without knowing the latitude, by observing 
the greatest and least meridian altitudes of the sun, and taking 
half the difference. This is the method practiced in ancient 
times by Hipparchus. (Art. 2.) On comparing observations 
made at different periods for more than two thousand years, it 
is found that the obliquity of the ecliptic is not constant, but 
that it undergoes a slight diminution from age to age, amount- 
ing to 52" in a century, or about half a second annually. We 



* Or, move properly, the tangents of the two circles (which denote the direc- 
tions of the curves at those points) are parallel. 

6 



82 



THE SUN. 



might apprehend that by successive approaches to each other 
the equator and ecliptic would finally coincide ; but astrono- 
mers have ascertained by an investigation, founded on the 
principles of universal gravitation, that this variation is con- 
fined within certain narrow limits,' and that the obliquity, 
after diminishing for some thousands of years, will then in- 
crease for a similar period, and will thus vibrate forever about 
a mean value. 

158. The dimensions of the earth's orbit, when compared 
with its own magnitude are immense. 

Since the distance of the earth from the sun is 95,000,000 
miles, and the length of the entire orbit nearly 600,000,000 
miles, it will be found, on calculation, that the earth moves 
1,640,000 miles per day, 68,000 miles per hour, 1,100 miles per 
minute, and nearly 19 miles every second, a velocity nearly 
fifty times as great as the maximum velocity of a cannon-ball. 
A place on the earth's equator turns, in the diurnal revolution, 
at the rate of about 1,000 miles an hour, and T 5 g of a mile per 
second. The motion around the sun, therefore, is nearly 70 
times as swift as the greatest motion around the axis. 

THE SEASONS. 

159. The change of seasons depends on two causes, (1) the 
obliquity of the ecliptic, and (2) the earth's axis always remain- 
ing parallel to itself. Had the earth's axis been perpendicular 
to the plane of its orbit, the equator would have coincided 
with the ecliptic, and the sun would have constantly appeared 
in the equator. To the inhabitants of the equatorial regions, 
the sun would always have appeared to move in the prime 
vertical ; and to the inhabitants of either pole, he would always 
have been in the horizon. But the axis being turned out of a 
perpendicular direction 23° 28', the equator is turned the same 
distance out of the ecliptic ; and since the equator and ecliptic 
are two great circles which cut each other in two opposite 
points, the sun, while performing his circuit in the ecliptic, 
must, evidently, be once a year in each of those points, and 
must depart from the equator of the heavens to a distance on 
either side equal to the inclination of the two circles ; that is, 
23° 28'. (Art. 22.) 



THE SEASONS. 



83 



160. The earth being a globe, the sun constantly enlightens 
the half next to him,* while the other half is in darkness. 
The boundary between the enlightened and the unenlightened 
part is called the circle of illumination. When the earth is at 
one of the equinoxes, the sun is at the other, and the circle of 
illumination passes through both the poles. When the earth 
reaches one of the tropics, the sun being at the other, the circle 
of illumination cuts the earth so as to pass 23° 28' beyond the 
nearer, and the same distance short of the remoter pole. These 
results would not be uniform, were not the earth's axis always 
to remain parallel to itself. The following figure will illustrate 
the foregoing statements. 

Fig. 31. 




* In fact, the sun enlightens a little more than half the earth, since, on ac- 
count of his vast magnitude, the tangents drawn from the sides of the sun to 
corresponding sides of th« earth, converge to a point behind the earth, as will be 
seen by and by, in the representation of eclipses. The amount of illumination, 
also, is increased by refraction. 



8i THE SUN. 

Let ABCD represent the earth's place in different parts of 
its orbit, having the snn in the center. Let A, C, be the po- 
sition of the earth at the equinoxes, and B, D, its positions at 
the tropics, the axis ns being always parallel to itself.* At A 
and C the snn shines on both n and s ; and now let the globe 
be turned round on its axis, and the learner will easily con- 
ceive that the snn will appear to describe the equator, which 
being bisected by the horizon of every place, of course the day 
and night will be equal in all parts of the globe. f Again, at 
B, when the earth is at the southern tropic, the sun shines 23^- a 
beyond the north pole n 9 and falls the same distance short of 
the south pole s. The case is exactly reversed when the earth 
is at the northern tropic and the sun at the southern. While 
the earth is at one of the tropics, at B for example, let us con- 
ceive of it as turning on its axis, and we shall readily see that 
all that part of the earth which lies within the north polar 
circle will enjoy continual day, while that within the south 
polar circle will have continual night, and that all other places 
will have their days longer as they are nearer to the enlight- 
ened pole, and shorter as they are nearer to the unenlightened 
pole. This figure likewise shows the successive positions of 
the earth at different periods of the year, with respect to the 
signs, and what months correspond to particular signs. Thus 
the earth enters Libra and the sun Aries, on the 21st of March, 
and on the 21st of June the earth is just entering Capricorn 
and the sun Cancer. 

161. Had the axis of the earth been perpendicular to the 
plane of the ecliptic, then the sun would always have appeared 
to move in the equator, the days would everywhere have been 
equal to the nights, and there could have been no change of 
seasons. On the other hand, had the inclination of the eclip- 
tic to the equator been much greater than it is, the vicissitudes 
of the seasons would have been proportionally greater than at 
present. Suppose, for instance, the equator had been at right 

° The learner will remark that the hemisphere toward n is above, and that 
toward s is below the plane of the paper. It is important to form a just con- 
ception of the position of the axis with respect to the plane of its orbit. 

f At the pole, the solar disk, at the time of the equinox, appears bisected by 
the horizon. 



85 

angles to the ecliptic, in which case, the poles of the earth 
would have been situated in the ecliptic itself; then in differ- 
ent parts of the earth the appearances would have been as 
follows : — To a spectator on the equator, the sun, as he left the 
vernal equinox would every day perform his diurnal revolution 
in a smaller and smaller circle, until he reached the north pole, 
when he would halt for a moment and then wheel about and 
return to the equator in the reverse order. The progress of 
the sun through the southern signs, to the south pole, would 
be similar to that already described. Such would be the ap- 
pearances to an inhabitant of the equatorial regions. To a 
spectator living in an oblique sphere, in our own latitude for 
example, the sun, while north of the equator, would advance 
continually northward, making his diurnal circuits in parallels 
further and further distant from the equator, until he reached 
the circle of perpetual apparition, after which he would climb 
by a spiral course to the north star, and then as rapidly return 
to the equator. By a similar progress southward, the sun 
would at length pass the circle of perpetual occultation, and 
for some time (which would be longer or shorter, according to 
the latitude of the place of observation) there would be contin- 
ual night. 

The great vicissitudes of heat and cold which would attend 
such a motion of the sun, would be wholly incompatible with 
the existence of either the animal or the vegetable kingdoms, 
and all terrestrial nature would be doomed to perpetual steril- 
ity and desolation. The happy provision which the Creator 
has made against such extreme vicissitudes, by confining the 
changes of the seasons within such narrow bounds, conspires 
with many other express arrangements in the economy of 
nature to secure the safety and comfort of the human race. 

FIG-URE OF THE EARTH'S ORBIT. 

162. Thus far we have taken the earth's orbit as a great 
circle, such being the projection of it on the celestial sphere ; 
but we now proceed to investigate its actual figure. 

Were the earth's path a circle, having the sun in the center, 
the sun would always be at the same distance from us; that is, 
the radius vector (the name given to a line drawn from the 



86 



THE SUN. 



center of the sun to the orbit of any planet) would always be 
of the same length. But the earth 's distance from the sun is 
constantly varying, which shows that its orbit is not a circle, 
having the sun at the center. We learn the true figure of the 
orbit, by ascertaining the relative distances of the earth from the 
sun at various periods of the year. When these are laid down 
according to their relative length, and making angles with 
each other equal to the changes in the sun's angular motions, 
a curve joining the extremities of these lines gives us our first 
idea of the shape of the orbit, which is found to be an ellipse. 
Thus (considering the earth E, for the present, as fixed, and 
the sun as the moving body), let E«, E&, Ec, &c. (Fig. 32), be 
the successive distances of the sun, laid down as just described ; 
then will the dotted line qfmt, which passes through their ex- 
tremities, show the form of the apparent solar orbit, with the 
earth in one of its foci. 

Fig. 32. 




163. These relative distances may be found by observing 
the changes in the sun's apparent diameter. Were the varia- 
tions in the sun's horizontal parallax considerable, as is the 
case with the moon's, this might be made the measure of the 
relative distances, for the parallax varies inversely as the dis- 
tance (Art. 82) ; but the whole horizontal parallax of the sun 
is only 9", and its variations are too slight and delicate, and 
too difficult to be found, to serve as a criterion of the changes 
in the sun's distance from the earth. But the changes in the 
surfs apparent diameter are much more sensible, and furnish 



FIGURE OF THE KARTll's ORBIT 



87 



a better method of measuring the relative distances of the 
earth from the sun. By a principle in optics, the apparent 
diameter of an object, at different distances from the spectator, 
is inversely as the distance.* Hence, the lines E«, E&, &c. 
(Fig. 32), are to be drawn proportional to the reciprocals of the 
apparent diameters of the sun. 

164. The point where the earth, or any planet, in its rev- 
olution, is nearest the sun, is called its perihelion / the point 
where it is furthest from the sun, its aphelion. The place of 
the earth's perihelion is known, since there the apparent mag- 
nitude of the sun is greatest ; and when the sun's magnitude is 
least, the earth is known to be at its aphelion. The sun's ap- 
parent diameter when greatest is 32' 35''. 6 ; and when least, 
31' 31"; hence the radius vector at the aphelion : rad vector 
at the perihelion : : 32.5933 : 31.5167 : : 1.034 : 1. Half of the 
difference of the two is equal to the distance of the focus of the 
ellipse from the center, a quantity which is always taken as 
the measure of the eccentricity of a planetary orbit. From 
twenty-four observations made with the greatest care by Dr. 
Maskelyne, at the Royal Observatory of Greenwich, the fol- 
lowing distances of the earth from the sun are determined for 
each month in the year. 



Time of Observation. 


Distances. 


Time of Observation. 


Distances. 


January 


12-13, 


0.98448 


July 


18-19, 


1.01658 


February 


17-18, 


0.98950 


August 


26-27, 


1.01042 


March 


14-15, 


0.99622 


September 


22-23, 


1.00283 


April 


28-29, 


1.00800 


October 


24-25, 


0.99303 


May 


15-16, 


1.01234 


November 


18-20, 


0.98746 


June 


17-18, 


1.01654 


December 


17-18, 


0.98415 



165. Having determined the form of the solar orbit, we 
are prepared to see what relation exists between the sun's 
angular velocity in this orbit, and the length of the radius 
vector. It has been already noticed (Art. 105), that the sun's 
progress in the ecliptic is fastest near the perihelion, and slow- 
est near the aphelion. For instance, the sun at perihelion ad- 

s More exactly, the* tangent of the apparent diameter is inversely as the dis- 
tance ; but in small angles like those concerned in the present inquiry, the angle 
and its tangent vary alike. 



SS THE SUN. 

vances about 61 ' in 2± hours, and at aphelion only about 57'. 

Now — = 1.07, which is the square of 1.034, the ratio of ap- 

o i 

parent diameters at the same points. Indeed, a careful com- 
parison of the sun's angular velocities, in all parts of the orbit, 
shows that they vary inversely as the squares of the distances. 
If changes in angular (i. <?., apparent) velocity were caused 
wholly by difference of distance, then it would vary inversely 
only as the first power of the distance, just as the apparent 
diameter does. But since the angular velocity varies inversely 
as the square, instead of the first power of the distance, the 
absolute velocity must also be greater as the distance is less, 
and vice versa. Thus we perceive, that when the sun is near- 
est to us, he appears to move fastest for two reasons, — first, be- 
cause the same rate of motion would appear greater, if nearer 
to us, and secondly, because the actual motion is then greater; 
and each of these is in the inverse ratio of the distance. 



166. It must be remembered, that this reasoning proceeds 
on the ground that the line of motion is everywhere at right 
angles to the radius vector. That this is true without sensible 
deviation, appears from the fact that the solar orbit is very 
nearly circular, with the earth at its center. If truly repre- 
sented on paper, it could not be distinguished by the eye from 
a circle. 

This relation between distances and apparent velocities 
having been once established, advantage may be taken of the 
rapid rate of change in the latter, to determine the variations 
in the radius vector more accurate- 
ly than can be done by the appar- 
ent diameter. 

167. The angular velocity being 
inversely as the square of the dis- 
tance in all parts of the solar orbit, 
it follows that the product of the 
angle described in any given time, 
by the square of the distance, is 
always the same constant quantity. 
For if of two factors, A xB, A is in- 



Fig. sa. 




89 

creased as B is diminished, the product of A and B is always 
the same. If, therefore, from the sun S (Fig. 33), two radii be 
drawn to T, B, the extremities of any. small arc, as that de- 
scribed in one day, and the angle between them be called S, 
then SB 2 xS gives the same constant product in all parts of the 
orbit. 

168. The radius vector of the solar orbit describes equal 
areas in equal times j and in unequal times, areas proportional . 
to the times. 

The solar orbit is so nearly a circle, that TB may everywhere 
be regarded as perpendicular to the radius SB or ST. Hence, 
the sector described in a given time, as one day, TSB oo SB x 
TB. But a circular arc varies both as the angle which it sub- 
tends, and also as the radius by which it is described ; there- 
fore TB oo SBxS. Hence, TSB oo SB 2 xS. But (Art. 167) 
this is a constant product; therefore, the area TSB is also con- 
stant, and the radius vector describes equal areas in equal 
times. 

The sun's orbit may be accurately represented by taking 
some point, as the perihelion, drawing the radius vector to that 
point, and, considering this line as unity, drawing other radii 
making angles with each other such that the included areas 
shall be proportional to the times, and of a length required by 
the distance of each point as given in the table (Art. 164). On 
connecting these radii, we shall thus see at once how little the 
earth's orbit departs from a perfect circle. Small as the differ- 
ence appears between the greatest and least distances, yet it 
amounts to nearly Jg of the perihelion distance, a quantity no 
less than 3,000,000 of miles. 

169. The foregoing method of determining the iigure of 
the earth's orbit is founded on observation / but this figure is 
subject to numerous irregularities, the nature of which cannot 
be clearly understood without a knowledge of the leading- 
principles of Universal Gravitation. An acquaintance with 
these will also be indispensable to our understanding the cause* 
of the numerous irregularities, which (as will hereafter appear) 
attend the motions of the moon and planets. To the laws of 
universal gravitation, therefore, let us next apply our attention. 



CHAPTEE III. 

OF CENTRAL FORCES — GRAVITATION. 

170. When a body moves in a curve of any kind, we rec- 
ognize the effect of two forces: one, an impulse, which acting 
alone would have caused a uniform motion in a straight line, 
and whose influence is always retained in the curve-motion ; 
the other, an accelerating force, which continually urges the 
body toward some point out of the original line of motion. 
The first is called the projectile force, the other the centripetal 
force. If the action of the latter were to cease at any moment, 
the body by its inertia would from that moment continue uni- 
formly in the direction in which it was then moving. Such 
motion in the tangent may be regarded as the effect of an im- 
pulse first given in the direction of that tangent. This supposed 
impulse is the projectile force for the moment in question ; but 
it is in truth the resultant of the original impulse, and the infi- 
nite series of actions already produced by the centripetal force. 

The centripetal force is of necessity infinitely small compared 
with the projectile force. For, if not, the curve would depart 
by a finite angle from the tangent ; whereas, by the very 
nature of the relation of a curve to its tangent, the angle is in- 
finitely small ; therefore, the deflecting force is infinitely small. 
But it produces finite deflection after a time, because its action 
is incessantly repeated. 

171. From a long and laborious examination of the record- 
ed observations of Tycho Brahe, Kepler deduced three laws 
relating to the movements of the planets, which are therefore 
called Kepler's laws. 

1. The orbit of every planet is an ellipse, having the sun in 
one focus. 

2. The radius vector of each orbit describes equal areas in 
equal times. 

3. The squares of the periodic times of the several planets 
vary as the cubes of their mean distances. 



CENTRAL FORCES. 91 

These were thus ascertained as facts, many years before 
Newton demonstrated, by mathematical reasoning, that they 
are necessarily involved in the laws of inertia and gravitation. 

The fundamental principles of all mechanical action, per- 
taining alike to terrestrial bodies and to the worlds scattered 
throughout space, are the following : 

1. Matter, until acted on by extraneous force, will remain 
perpetually in its present condition, whether of rest or straight 
uniform motion. 

2. All motions communicated to a body coexist in the mo- 
tion of the body. 

3. To every action there is an equal and opposite reaction. 

4. All masses tend toward each other, with a force varying 
directly as the quantity of matter, and inversely as the square 
of the distance. 

The three first, the laws of inertia, of coexistent motions, and 
of equal action and reactio?i, were seen to be the true first 
principles in the Mechanics of terrestrial bodies. But they are 
equally essential in Astronomy — the celestial Mechanics ; and 
not only does no fact in this science militate against them, but, 
on the contrary, they form the basis of all correct reasoning on 
the motions of the heavenly bodies. The fourth, usually called 
the law of gravity, is far more prominent in Astronomy than 
in Mechanics, but harmonizes with all the facts of both. We 
proceed to show that Kepler's laws and other laws of central 
forces, are the necessary consequences of the above-named me- 
chanical principles. 

172. Whatever path a body describes under the influence 
of a projectile and a centripetal force, the radius vector of that 
path passes over equal spaces in equal times. 

Let S (Fig. 34) be the center of attraction, and suppose the 
projectile force in the line YK. to be such as to cause the body 
to pass over the equal spaces PQ, QE,, &c, each in a certain 
unit of time. When the body reaches Q, let the action toward 
S be sufficient to move it over QY in the same time in which 
by the original impulse it would describe QK. Then it will 
in the same time describe the diagonal QC of the parallelo- 
gram. Join ES and CS. The triangles QSC and QSXv are 
equal; but QSR=QSP; .-. QSC=QSP. That is, the areas do- 



92 



UNIVERSAL GRAVITATION. 
Fig. 34. 




scribed in the first and second units of time are equal. In like 
manner, by supposing a second action toward S to occur at C, 
a third at D, &c, it is proved that QCS, CDS, DES, &c, 
which are described in equal times, are equal. This is true, 
however small the unit of time between the successive actions 
toward S, and is therefore true, wdien the central force acts in- 
cessantly^ and causes curvilinear motion. As the diagonal of 
each parallelogram is in the same plane with its two sides, it 
is obvious that the whole orbit lies in one and the same plane. 

173. Conversely, if equal areas be described about a point 
in equal times, by the radius vector, the deflecting force acts 
toward that point. For PSQ=QSR, as before (Fig. 34); 
and by supposition, PSQ=QSC ; .\ QSC=QSR; hence CR is 
parallel to QS, and QC is the diagonal of a parallelogram, 
whose side QV, in which the deflecting force acts, is directed 
toward S. 

It has been shown (Art. 168) that, from observations on the 
angular velocity of the earth about the sun, equal areas are in 
fact described by the radius vector in equal times. It is there- 
fore inferred that there is an accelerating force urging the 
earth toward the sun. 

174. The velocity at any point of an orbit, varies inversely 



CENTRAL FORCES. 93 

as the perpendicular from the center of force to the tangent at 
that point. 

Let SY (Fig. 31) be perpendicular to PQ ; then the area 

SPQ=iPQxSY, which varies as PQxSY ; .-. PQ oo -?Q. 

b X 

But PQ oo Y, the velocity at P ; and the area SPQ is constant ; 
.-. Too -~^ 3 or the velocity varies inversely as the perpen- 

dicular from S upon the line in which the body is moving ; 
in other words, upon the tangent of its path, if it describes 
a curve. 

In the orbits of the planets, since they are very nearly cir- 
cular, SY meets the path almost at the point where the body 
is moving, and therefore is about equal to the radius vector ; 
so that in the planetary orbits, the absolute velocity varies in- 
versely as the radius vector very nearly. We have already 
noticed this to be sensibly true in the case of the earth's orbit. 
(Arts. 165, 166.) It follows from the above reasoning, that in 
a circular orbit, where the radius vector is constant, the ve- 
locity of the body is uniform. 

175. When a body moves in a curve, since by its inertia it 
tends at each point to proceed in the tangent at that point, 
there is a continual outward pressure directed from the center 
of force; this is called the centrifugal force. It may be regard- 
ed as that infinitesimal component of the projectile force, which 
opposes the action of the centripetal, the motion along the curve 
being the other component. If the body is maintained at the 
same distance from the center (that is, in the circumference of a 
circle), the centrifugal force equals the centripetal ; but in orbits 
of other forms, it is sometimes greater and sometimes less than 
the centripetal. 

17G. In a circular orbit, the central force (either centripe- 
tal or centrifugal) varies as the square of the velocity divided by 
the, radius. * 

If v~ the uniform velocity in the orbit, and £=the infin- 
itely small portion of time of describing the minute arc Ab, 
and r=the radius of the circle, then Ab~vt. But A5, or its 




94: UNIVERSAL GRAVITATION. 

chord, is a mean proportional between its 
versed sine Aa, and the diameter 2r ; or 

~KF 2 vH 2 

Aa=— — = — — , which, in a given time, varies 

v 2 . . 

as — , that is (since the central forced is meas- 

v 2 
nred by Act), f <*>—< 

Hence, in a given circle, where r is constant, the central 
force varies as the square of the velocity. In whirling a ball, 
for instance, with a string of given length, if the velocity is 
doubled, the strain upon the string (the centrifugal force) is 
four times as great, and the strength of the string (the centrip- 
etal force) needs also to be four times as great. If a train of 
cars goes round a curve with a velocity 1£ times that which is 
intended, its tendency to be thrown from the track is increased 
2£ times. 

177. In a circular orbit, the central force (centripetal or 
centrifugal) varies as the radius of the circle divided by the 
square of the time of revolution. 

Let £=the time of describing the whole circumference 2nr ; 

QTcr r r 2 

.". 2-rrr=vt, and #=— — •' which varies as--; .*. v oo -. But (Art. 

t to" 

v 2 r 2 r r 

176) fa — oo -i-rroo^; .-.fee—. Hence, if the periodic time 
r z t t 

ls the same, the attraction to the center must be increased in 

the same ratio as the radius of the orbit, for then fco r. If a 

string is twice as long, it must have twice the strength, in order 

to whirl a ball at the same rate of revolution. 

178. If a body describes an elliptical orbit by a centripetal 
force which acts toward the focus, that force varies inversely 
jts the square of the distance. 

Let the body be at M (Fig. 36), and MF the radius vector 
at that point. Let MO be the radius of curvature at M, and 
therefore perpendicular to the tangent ; and suppose MN to 
be an infinitely small arc described in a given small portion of 
time. Draw FP perpendicular to the tangent MP, NK to 



CENTRAL FORCES. 



95 



FM, and IH to MO ; then PFM, MHI, KOT, are similar tri- 
angles. MJST, considered as a straight line, is described by the 
joint action of the centripetal force MI, and the projectile 
force, which is equal and parallel to IN. The motion in MJ 



Fig. 36. 




may be regarded as uniformly accelerated, because in the in- 
finitely small time of describing it, the centripetal force may 
be considered constant. Hence, 2MI may be taken as the 
measure of the centripetal force f.* Therefore fee ML It is 

to be proved that MI go ^-r . 
r FM 2 



179. By similar triangles, MI : MH : : 1STI : NK ; 

,.mi^mh|I. 

Now, the chord MN" is a mean proportional between the 

ME" 2 
versed sine MH, and the diameter 2MO ; or IH=- TF7 ; ; 
' ' 2MO ' 

NH 2 

but, as the arc is infinitely small, KH=MN ; .*. MH=-^. 
' J J* ' 2MO 

Again, the versed sine MH, and therefore HI, is infinitely 

small compared with E"H, and NI may be substituted for 

NH; 

KI 2 

••• MH =iMO" 



Nat. Phil., Art. 28. 



UNIVERSAL GRAVITATK >N. 



Now, it is shown in conic sections, that MO= --( v^p J > •'• by 

£>/NI \ 3 
similar triangles, MO=M ^p J . Substituting this for MO 

NK 3 

in the equation for MH above, we have MH = — ==. Hence, 

in the equation for MI, we have 

„ NK 3 NI l x _, 
MI= — = TT x ==- = -NK'. 

J9 . NI NK ^ 

Now the sector FMN is measured bv fFM.NK; .*. NK= 
2FMN , ATir2 4FMN 2 _- T 4FMN 2 _ ' 

~fm" ' and "TiF~ ; •'• ml= J7fW ' as equal 

areas are described by the radius vector in equal times, FMN 
is constant. Therefore 

m (=/)».j5p; 

that is the centripetal force varies inversely as the square of 
the distance. 

180. It is thus proved, that in any elliptical orbit de- 
scribed about the focus as the center of attraction, the inten- 
sity of that attraction varies inversely as the square of the ra- 
dius vector. As there is nothing in the foregoing demonstra- 
tion to limit the conclusion to the orbits which are nearly cir- 
cular, like those of the planets, we are at liberty to apply it to 
orbits of extreme eccentricity, as those of the comets. And it 
is proved by Newton in his Principia, that the same law of 
force is necessary, in order that a body may describe any one 
of the conic sections about its focus as the center of attraction. 

181. And not only does this law prevail in all parts of any 

* Jackson's Conic Sections. The same may be derived from Coffin's Con. 

(FM MV) 3 

Sec. Pr. V., Curvature. E 2 or M0 2 = -r; a and b being the semi-axes ; .-. 

a?b 

1 1 A 1 

M0= X(FM.MV) 2 . Multiply by (6 2 ) 2 , and divide by its equal (FP.yL) 2 ; 
ab 

6 2 /FM MV\^ fi 2 /FM ! \l 
then M0= ( ' - l 2 =-f ~„g l 2 , since FMP and VML, are similar. But 

° 3 P 1 • *u * *™ J°/FM 2 \! »/FM\3 
-—-. p being the parameter ; .-. M0 = -( ) 2 =- I — J 



CENTRAL FORCES. 97 

one orbit, but it is true also, that all the different bodies of a 
system, describing orbits about the same center of force, are 
urged toward that center by attractions which vary, from one 
orbit to another, inversely as the square of the distance. 

Let a be the semi-major, and b the semi-minor axis of any 
elliptic orbit. Then a is the mean distance of all points of the 
orbit from the focus. By a rule of mensuration, the area of 
the ellipse =7rah. Ifs=the area described by the radius vec- 
tor in a unit of time, as one second, and £=the number of sec- 
onds in the whole period of revolution, then the ellipse also=&9. 

Therefore nab~ts ; and t— ; and t 2 = — j—. By Kepler's 

a 2 b 2 b 2 

third law (Art. 171), t 2 ooa s ', .*. — r ooa 3 ; ,\ -aos 2 . But, be- 

S & 

cause the semi-parameter ^-, is a third proportional to the 

h 2 7) 7) 7) 

semi-axes a and b, — =^; .*. ^<x>s 2 . Hence, substituting -^ 

for s\ that is, FMN 2 , in the equation for MI (Art. 179), we 

4FMN 2 2» 2 1 

&ndm ^^¥W=^W = FW ; -'-^YW 0r ' the force 
varies inversely as the square of the distance, in different or- 
bits, as well as in different parts of the same orbit. 

The satellites which revolve about the planets, are found to 
conform to Kepler's laws, and therefore the force which urges 
them toward their respective primaries, varies in each case in- 
versely as the square of the distance. 

182. But the inquiry still remains, does the law of gravity, 
as demonstrated in the foregoing articles, hold good at the 
smallest distances also ? For example, do the tendencies of 
bodies resting on the earth, and of those elevated in the air, 
and of the moon, toward the earth's center, come under the 
same general law ? This is the very question which presented 
itself to the mind of Newton, after he had discovered that the 
force which deflects the planets from their lines of motion 
toward the sun, varies inversely as the square of their distance 
from it. As he noticed the fall of an apple, the inquiry arose, 
may not this/M be of the same nature as the bending of the 

7 



98 



UNIVERSAL GRAVITATION". 




moon's path toward the earth, and may not the force in the 
two cases be as the squares of the distances inversely ? 

Let us then find what is the distance through which the 
moon actually descends in a second of time. Let the earth be 
at E (Fig. 37), and Ah the arc described by the moon in one 
second. As she was going toward B 
at the point A, she would have gone 
over the line AB in one second, if 
some influence had not turned her 
aside. This influence must be direct- 
ed toioard the earth E, because it is 
around E that the radius vector de- 
scribes equal areas in equal times 
(Art, 173). Therefore B6, or the 
versed sine Act (which may be con- 
sidered equal to it), is the distance 
fallen through in one second. Now 
the distance of the moon from the 
earth's center is 238,545 miles (Art. 
201). Hence, the circumference is known. The time of rev- 
olution is 27.32 days. (Art. 213.) Therefore Ah, the distance 
traveled in one second is obtained. This is so small, that its 
versed sine Aa can not be calculated by ordinary trigonomet- 
rical tables ; but is easily and accurately determined by ge- 
ometry ; thus, 2AE : Ah : : Ah : Ad ; since the arc Ah and the 
chord Ah may be considered identical. A a, thus calculated, 
is found to be 0.0535 of an inch. At the surface of the earth, 
a body falls about 16^2 feet in the first second. In order to 
diminish this for the moon's distance, we make the proportion, 
(238.515) 2 : (3956) 2 : : 16 T2 - ft, : 0.05S6, agreeing very accu- 
rately with the distance which the moon actually falls from a 
tangent in a second of time. When Newton, however, first 
made the comparison just described, the result was quite un- 
satisfactory. The tendency of the moon was about £ too 
great, Near 20 years afterward, when a new measurement of 
a degree of the meridian had been made, and thus a corrected 
magnitude of the earth, he repeated the process, and found 
the law of attraction in this case to be the same as elsewhere 
in the solar system. 

Again, the numerous disturbances which the bodies of the 



CENTRAL. FORCES. 99 

solar system produce in each other's motions, are all accounted 
for by applying the same law. If a planet moves for a time 
toward another, it is accelerated ; and its acceleration is 
greater, as the square of the distance is less, and it is retarded 
according to the same law, when departing from it. 

Since all bodies manifest a tendency toward each other, it 
is natural to suppose that the tendency should vary as the 
quantity of matter, other things being equal. Both observa- 
tion and calculation confirm, without any exceptions, such a 
supposition; and therefore a full statement of the law of 
gravity includes the fact, that it varies directly as the quanti- 
ty of matter. 

183. Since, as we have seen, gravity just at the earth's 
surface is governed by the same law of distance as it is further 
off, therefore the curved paths of projectiles are of the same 
nature as the orbits of planets and satellites ; that is, they are 
ellipses, one of whose foci is at the earth's center. And there 
is no real discrepancy between this statement and that proved 
in Mechanics,* that the path of a projectile is a parabola* In 
that demonstration, it was assumed that the lines in which the 
gravitating force acts at each point of the path, are parallel to 
each other, and that the force is constant, neither of which is 
strictly true ; since the verticals all meet at the center of at- 
traction, and the intensity of gravity slightly increases as the 
body descends. Knowing the distance and period of the 
moon, it is easy to find by Kepler's third law the period of 
revolution in case of a given projectile, if its orbit could be 
completed in accordance with the law. Any force, which man 
could apply, would carry the perigee so little beyond the cen- 
ter of the earth, that the mean distance -might be called one- 
half the radius of the earth. Therefore, calling the moon's 
distance 60 radii, and her period 27} days, we should have 
(60) 3 : (if : : (27J-) 2 : x\ the square root of which would be 
about 30 minutes. Every projectile, then, if it were free to 
complete its orbit, unobstructed, and according to the law of 
gravity which prevails outside of the earth, would make an 
entire revolution, and return to its place in half an hour. Ae - 

* Nat. Phil., Art. 49. 

toffc 



100 



UNIVERSAL GRAVITATION. 



cording to Art, 174, the velocity at the perihelion would be as 
much greater than that of projection as its distance from the 
center is less. 

Fig. 38. 




184. In Fig. 38, let AC be the vertical, and AB, at right 
angles to it, the line of projection; then, according to the 
velocity given, AD, AE, or AF might be the commencement 
of the orbit; from the point A the velocity would increase 
continually till the moment of nearest approach to the center, 
from which point it would decrease all the way back to A. In 
all these orbits, the center of the earth will occupy the focus 
most remote from the point of projection ; in other words, the 
place of projection is the apogee of the orbit. Now suppose 




tnat the velocity should be greatly augmented, as in Fig. 39, 
so that, in one case, the curve should strike the earth at D, in 
another at E, and so on, until finally a force should be given 
sufficient to carry the body round in a circle. This last case 



CENTRAL -FORCES. 101 

will happen when the centrifugal force has been increased so 
as to just equal gravity (Art. 175). As the mean distance 
now equals the radius of the earth, the time of revolution is 
easily found to be lh. 24m. 39s. Any increase of projectile 
force beyond this will again produce an ellipse as PK, whose 
perigee is at P ; and we can imagine the velocity of projection 
increased until the ellipse becomes one of extreme eccentricity, 
and then changes into a parabola, and then into an hyperbola, 
in which last cases the body will never return, or even reach 
the point of apogee. 

185. If we suppose the projectile motion of the earth or 
any other planet, to have been produced by a single impulse, 
that impulse may also have caused the diurnal rotation of the 
body. If it had been directed in a line passing through the 
center of gravity of the planet, then it would have caused a 
progressive motion, without rotation on the axis. But if the 
line of the impulse did not pass through the center of gravity, 
then besides the motion forward, there would also be a rota- 
tion whose velocity would depend on the distance of the line 
from the center. According to the calculation made by Ber- 
nouilli, the earth's progressive and rotary motions might have 
both been produced by the application of a force in a line 
passing twenty-four miles from the center of the earth, on the 
side most remote from the sun.* 

Had it been directed through a point lying on the side near- 
est the sun, the diurnal rotation would obviously have been 
retrograde. 

186. But if a force were applied to a planet as we have 
been supposing, what effect would be produced on the system 
as a whole ? To simplify the case, suppose the sun and one 
planet at rest, and at a given distance apart. Let them begin 
to attract each other, and at the same instant let an impulse be 
applied to the planet at right angles to the line joining the two, 
and of sufficient intensity to cause revolution in a circle ; will 
that circle be described about the sun as a stationary body ? 
It will not, but around the common center of gravity as a mov- 

8 Francceur, Uian,, p. 49. 



102 



TJNIVEKSAL GRAVITATION. 



ing point, while the sun will do the same in consequence of the 
mutual attraction between them. Figure 40 will illustrate the 
case. It is proved (Nat. Phil., Art. 89) that a force applied to 
one body of a system will produce the same effect on the cen- 
ter of gravity of that system, as if the force were applied to 

Fig. 40 



A g 




r^Os 



the entire system collected at that center. Therefore, let S be 
the sun, E the earth, and C their center of gravity ; and let the 
impulse be such as to move the sum of both bodies over Ca, 
ah, he, &c, in any equal times, while in each unit of time it 
carries E over a given arc, say 45° of its circle, then, when the 
center is at a, E is at 1, 45° from a perpendicular at a. But S 
must be on the opposite side of a, and at the same distance from 
it as from C before, because E's distance from the center re- 
mains unaltered. Therefore, by the impulse given to E, and 
the mutual attraction between E and S, the latter has been 
drawn along from S to 1'. Drawing again two circles from 
the center b, one for E, the other for S, the next positions for 
the bodies are 2 and 2'. While E was on the upper side of Cd, 
S was drawn toward that line, and now crosses it, and by its 
own inertia continues upward, although E is now below the 
line. In this manner the two bodies revolve about a moving 
center, describing circles relatively to that, but curves of a to- 
tally different character in space. These curves are always 
some variety or other of the curve called an epicycloid. In the 



CENTRAL FORCES. 



103 



case represented in the figure, the smaller body describes an 
epicycloid which forms a series of loops, intersecting its own 
path at every revolution, while the path of the heavier body is 
of a waving form. The body E retrogrades on the lower part 
of the loop, from 3 to 5, while S advances continually, but 
with unequal velocities, each body being alternately drawn 
forward and held back by the other. 

The only way in which two separate bodies could be made 
to rotate about a fixed center of gravity, would be to give an 
equal impulse to each body, and in opposite directions. Two 
such forces would constitute a couple £N"at. Phil., Art. 57), 
whose effect is to produce rotation merely. 

187. The learner may find some difficulty in understanding 
why a planet, when it reaches its aphelion C (Fig. 41), where 
gravity is more feeble than it had 
previously been, should begin from 
that point to approach the sun ; and 
again, why, at the perihelion G, 
where gravity is greatest, it should 
begin to depart, instead of approach- 
ing nearer and nearer in a spiral, till R 
it falls upon the sun. This is owing 
to the change in velocity, and conse- 
quently in centrifugal force. As the 
planet ascends through H, K, and 
A, the action is partly against its 
motion, and consequently retards it, 
till the projectile force is too small to maintain it at that dis- 
tance, and it begins from C to approach. In approaching, the 
attraction of the sun partly conspires with its own inertia, to 
accelerate it, through D, E, and F ; its velocity is thus increased, 
till the centrifugal force becomes so great at G, precisely op- 
posite to C, that it commences once more to depart in its former 
path. Of course these effects could not take place unless the 
centrifugal force varied more rapidly than the centripetal, 
which is true ; for it is proved that the centrifugal force in an 
orbit varies inversely as the cube of the distance, while the cen- 
tripetal varies only as the square inversely.* 




* M. Stewart's " Phys. and Math. Tracts," Prop. 8. 



CHAPTEE IT. 

PRECESSION OF THE EQUINOXES — NUTATION ABERRATION — MOTION 

OF THE APSIDES MEAN AND TRUE PLACES OF THE SUN. 

188. The Precession of the Equinoxes is a slow out con- 
tinual shifting of the equinoctial points from east to west. 

Suppose that we mark the exact place in the heavens where 
the sun crosses the equator, the present year in March, and 
that this point is close to a certain star ; next year the sun will 
cross the equator a little way westward of that star, and so 
every year a little further westward, until, in a long course of 
ages, the place of the equinox will occupy successively every 
part of the ecliptic, until we come round to the same star 
again. As, therefore, the sun, revolving from west to east in 
his apparent orbit, comes round toward the point where it 
left the equinox, it meets the equinox before it reaches that 
point. As the time of crossing, in every instance, precedes the 
time on the previous year, the phenomenon is called the Pre- 
cession of the Equinoxes, and the fact is expressed by saying 
that the equinoxes retrograde on the ecliptic, until the line of 
the equinoxes makes a complete revolution from east to west. 
The equator is conceived as sliding westward on the ecliptic, 
always preserving the same inclination to it, as a ring placed 
at a small angle with another of nearly the same size, which 
remains fixed, may be slid quite around it, giving a corre- 
sponding motion to the two points of intersection. It must be 
observed, however, that this mode of conceiving of the pre- 
cession of the equinoxes is purely imaginary, and is employed 
merely for the convenience of representation. 

189. The amount of precession annually is 50".l ; whence, 
since there are 3600" in a degree, and 360° in the w T hole cir- 
cumference, and consequently, 1 296000", this sum divided by 
50.1 gives 25868 years for the period of a complete revolution 
of the equinoxes. 



PRECESSION OF THE EQUINOXES. 105 

190, Suppose now we fix to the center of each of the two 
rings (Art. 188} a wire representing its axis, one corresponding 
to the axis of the ecliptic, the other to that of the equator, the 
extremity of each being the pole of its circle. As the ring de- 
noting the equator turns round on the ecliptic, which with its 
axis remains fixed, it is easy to conceive that the axis of the 
equator revolves around that of the ecliptic, and the pole of 
the equator around the pole of the ecliptic, and constantly at a 
distance equal to the inclination of the two circles. To trans- 
fer our conceptions to the celestial sphere, we may easily see 
that the axis of the diurnal sphere (that of the earth produced, 
Art. 28) would not have its pole constantly in the same place 
among the stars, but that this pole would perform a slow rev- 
olution around the pole of the ecliptic from east to west, com- 
pleting the circuit in about 26,000 years. Hence the star 
which we now call the pole-star, has not always enjoyed that 
distinction, nor will it always enjoy it hereafter. When the 
earliest catalogues of the stars were made, this star was 12° 
from the pole. It is now 1° 24/, and will approach still near- 
er ; or, to speak more accurately, the pole will come still near- 
er to this star, after which it will leave it, and successively 
pass by others. In about 13,000 years, the bright star Lyra, 
which lies on the circle of revolution opposite to the present 
pole-star, will be within 5° of the pole, and will constitute the 
Pole-star. As Lyra now passes near our zenith, the learner 
might suppose that the change of position of the pole among 
the stars, would be attended with a change of altitude of the 
north pole above the horizon. This mistaken idea is one of 
the many misapprehensions which result from the habit of 
considering the horizon as a fixed circle in space. However 
the pole might shift its position in space, we should still be at 
the same distance from it, and our horizon would always reach 
the same distance beyond it. 

191. The precession of the equinoxes is an effect of the 
spheroidal figure of the earth, and arises from the attraction of 
the sun and moon upon the excess of matter about the earth's 
equator. 

Were the earth a perfect sphere, the attractions of the sun 
and moon upon the earth would be in equilibrium among 



106 THE SUN. 

themselves. But if a globe were cut out of the earth (taking 
half the polar diameter for radius), it would leave a protuber- 
ant mass of matter in the equatorial regions, which may be 
considered as all collected into a ring resting on the earth. 
The sun being in the ecliptic, while the plane of this ring is 
inclined to the ecliptic 23° 28', of course the action of the sun is 
oblique to the ring, and may be resolved into two forces, one 
in the plane of the equator, and the other perpendicular to it. 
The latter only can act as a disturbing force, and tending as 
it does to draw down the ring to the ecliptic, the ring would 
turn upon the line of the equinoxes as upon a hinge, and 
dragging the earth along with it, the equator would ultimately 
coincide with the ecliptic were it not for the revolution of the 

Fig. 42. 




earth upon its axis. Let EC (Fig. 42) be the ecliptic, and QE. 
the equator. Any particle A, of the ring, by its inertia of 
rotation, tends to move toward T in the plane QB. Let AB 
represent this force, and AF the pressure toward EC produced 
by the sun; then the resultant will be AD; shifting the 
equinox backward from T to T'. Each particle is subjected 
to this influence, except at the moment (each day) of crossing 
T and =~=, so long as the sun is not himself in the line T =s= pro- 
duced, which occurs in March and September. The effect is 
then interrupted for a short time. 

192. The moon conspires with the sun in producing the 
precession of the equinoxes, its effect, on account of its nearness 
to the earth, being more than double that of the sun, or as 7 
to 3. The planets likewise, by their attraction, produce a 



PRECESSION OF THE EQUINOXES. 107 

small effect upon the equatorial ring, but the result is slightly 
to diminish the amount of precession. The whole effect of the 
sun and moon being 50".41, that of the planets is 0".31, leaving 
the actual amount of precession 50". 1. 

The law of compositions of motion in rotation is analogous 
to that for the composition of rectilinear motions (Nat. Phil., 
Art. 12), and may be stated thus : if two forces are applied to 
a body, which, separately, would cause it to rotate on two dif- 
ferent axes, their joint action will produce rotation on an axis 
lying between the others, at angles whose sines are inversely as 
the forces. In the precession of equinoxes, the earth rotates on 
the diurnal axis by one force, and the sun tends to revolve it 
on an axis passing through the equinoxes. As the latter force 
is minute, compared with the other, the new axis is shifted by 
a very small angle each year from the axis of rotation toward 
the line of equinoxes. And as this line slides along by the 
same quantity, the two axes remain perpetually at right angles 
with each other.* 

193. The time occupied by the sun in passing from an 
equinox or solstice round to the same point again, is called the 
tropical year. As the sun does not perform a complete rev- 
olution in this interval, but falls short of it 50". 1, the tropical 
year is shorter than the sidereal by 20m. 20s. in mean solar 
time, this being the time of describing an arc of 50". 1 in the 
annual revolution. The changes produced by the precession 
of the equinoxes in the apparent places of the circumpolar 
stars, have led to some interesting results in chronology. In 
consequence of the retrograde motion of the equinoctial points, 
the signs of the ecliptic (Art. 35) do not correspond at present 
to the constellations which bear the same names, but lie about 
one whole sign, or 30°, westward of them. Thus, that division 
of the ecliptic which is called the sign Taurus, lies in the con- 
stellation Aries, and the sign Gemini in the constellation 
Taurus. Undoubtedly, however, when the ecliptic was thus 
first divided, and the divisions named, the several constella- 



f The precession of equinoxes, and other cases of compound rotations, are 
finely illustrated by the gyroscope of Foucault, and still better by Johnson's 
rotascope. 



108 THE SUN. 

tions lay in the respective divisions which bear their names. 
How long is it, then, since our zodiac was formed ? 

50".l : 1 year : : 30° (= 10S000") : 2155.6 years. 
The result indicates that the present divisions of the zodiac 
were made soon after the establishment of the Alexandrian 
school of astronomy. (Art. 2.) 

NUTATION. 

104. Nutation is a vibratoiy motion of the earth's axis, 
arising from periodical fluctuations in the obliquity of the 
ecliptic. 

If the sun and moon moved in the plane of the equator, 
there would be no precession, and the effect of their action in 
producing it varies with their distance from that plane. Twice 
a year, therefore, — namely, at the equinoxes, — the effect of the 
sun is nothing; while at the solstices the effect of the sun is a 
maximum. On this account the obliquity of the ecliptic is 
subject to a semi-annual variation, since the sun's force, which 
tends to produce a change in the obliquity, is variable, while 
the diurnal motion of the earth, which prevents the change 
from taking place, is constant. Hence the plane of the equa- 
tor is subject to an irregular motion, which is called the Solar 
Nidation. The name is derived from the oscillatory motion 
communicated by it to the earth's axis, while the pole of the 
equator is performing its revolution around the pole of the 
ecliptic (Art. 190). The effect of the sun, however, is less than 
that of the moon, in the ratio of 2 to 5. By the nutation 
alone, the pole of the earth would perforin a revolution in a 
very small ellipse, only 18" in diameter, the center being in 
the circle which the pole describes around the pole of the 
ecliptic ; but the combined effects of precession and nutation 
convert the circumference of this circle into a wavy line. The 
motion of the equator occasioned by nutation, causes it alter- 
nately to approach to and recede from the stars, and thus to 
change their declinations The solar nutation, depending on 
the position of the sun with respect to the equinoxes, passes 
through all its variations annually; but the lunar nutation, 
depending on the position of the moon, with respect to her 
nodes, varies through a period of about 18^- years. 



ABERRATION. 109 



ABERRATION. 




195. Aberration is an apparent change of place in the 
stars, occasioned by the joint effects of the motion of the earth 
in its orbit, and the progressive motion of light. 

Suppose the earth to move from C to E, while the light from 
S describes the line DE. If they arrive together at the point 
E, the impression on the eye will not be 
the same as if the observer had been at 
rest, but it will appear to come in the di- 
rection of S'E, the star being apparently 
thrown forward from S to S\ For, make 
EA = DE, and complete the parallelogram 
C A ; and suppose, according to the princi- 
ple of equal action and reaction, that the 
light has a motion EC given to it, in place 
of the earth's motion CE ; then the two 
motions EA and EC will produce the re- 
sultant EB, as though the light had come 
from S' instead of S. 

Since the earth moves 19 miles, and light 192,000 per second, 
if S is in a direction perpendicular to the line of the earth's 
motion, the right-angled triangle ECB gives about 20".5 for 
the displacement of the star. In fact, however, it was the ob- 
served displacement of 20". 5 in all stars situated 90° from the 
direction in which the earth is moving at any time, which led 
to the knowledge of the velocity of light ; thus, 

tan 20".5 : K : : 19 : 192,000. 

If there were no change in the aberration of a star, the fact 
of such aberration could never have been discovered. When 
we move directly toward or from a star, it plainly has no ab- 
erration ; but three months before that time our motion crosses 
the line of the rays at right angles, and also three months after, 
but in a contrary direction. Hence any star in the plane of 
the ecliptic apparently moves back and forth over an arc of 
41' (2 x 20". 5) in a year. Those out of the ecliptic seem to 
describe elliptic orbits, whose major axis is ll", and of various 
eccentricity, the minor axis increasing with their latitude. 



110 THE SUN. 



MOTION OF THE APSIDES. 

196. The two points of the ecliptic, where the earth is at 
the greatest and least distances from the sun respectively, do 
not always maintain the same places among the signs, but 
gradually shift their positions from west to east. If we accu- 
rately observe the place among the stars, where the earth is at 
the time of its perihelion the present year, we shall find that 
it will not be precisely at that point the next year when it 
arrives at its perihelion, but about 12" (11 " . 66) to the east of 
it. And since the equinox itself, from which longitude is reck- 
oned, moves in the opposite direction 50". 1 annually, the lon- 
gitude of the perihelion increases every year 61".76, or a little 
more than one minute. This fact is expressed by saying that 
the line of the apsides of the earth's orbit has a slow motion 
from west to east. Tt completes one entire revolution in its 
own plane in about 100,000 years (111,119). 

The mean longitude of the perihelion at the commencement 
of the present century was 99° 30' 5", and of course in the 
ninth degree of Cancer, a little past the winter solstice. In 
the year 1248, the perihelion was at the place of this solstice. 

The advance of apsides is caused by the attraction of the 
other planets. As their weight is mostly outside of the earth's 
orbit, their effect is to diminish the earth's tendency toward 
the sun. But this influence is, on the whole, greater when 
the earth is most distant; that is, at aphelion. Consequently 
the earth passes a little further onward than at the preceding 
revolution, before turning to approach the perihelion. Thus 
the aphelion advances, and the law of revolution requires that 
the perihelion be opposite to it ; hence that advances also. 

197. The angular distance of a body from its perihelion is 
called its Anomaly ; and the interval between the sun's pass- 
ing the point of the ecliptic corresponding to the earth's peri- 
helion, and returning to the same point again, is called the 
anomalistic year. This period must be a little longer than 
the sidereal year, since, in order to complete the anomalistic 
revolution, the sun must traverse an arc of 11".66 in addition 
to 360°. Now 360° : 365.256 : : 11".66 : 4m. 44s. 



MEAN AND TRUE PLACES OF THE SUN. Ill 

198. Since the points of the annual orbit, where the sun is 
at the greatest and least distances from the earth, change their 
position with respect to the solstices, a slow change is occa- 
sioned in the duration of the respective seasons. For, let the 
perihelion correspond to the place of the winter solstice, as 
was the case in the year 1248 ; then as the sun moves more 
rapidly in that part of his orbit, the winter season will be 
shorter than the summer. But, again, let the perihelion be at 
the summer solstice, as it will be in the year 11740,* then the 
sun will move most rapidly through the summer months, and 
the winters will be longer than the summers. At present the 
perihelion is so near the winter solstice, that, the year being 
divided into summer and winter by the equinoxes, the six 
winter months are passed over between seven and eight days 
sooner than the summer months. 

MEAN AND TRUE PLACES OF THE SUN. 

199. The Mean Motion of any body revolving in an orbit, 
is that which it would have if, in the same time, it revolved 
uniformly in a circle. 

In surveying an irregular field, it is common first to strike 
out some regular figure, as a square or a parallelogram, by run- 
ning long lines, and disregarding many small irregularities in 
the boundaries of the field. By this process, we obtain an ap- 
proximation to the contents of the field, although we have per- 
haps thrown out several small portions which belong to it, and 
included a number of others which do not belong to it. These 
being separately estimated and added to or subtracted from 
our first computation, we obtain the true area of the field. In 
a similar manner, we proceed in finding the place of a heaven- 
ly body, which moves in an orbit more or less irregular. Thus 
we estimate the sun's distance from the vernal equinox for every 
day of the year at noon, on the supposition that he moves uni- 
formly in a circular orbit : This is the sun's mean longitude. 
We then apply to this result various corrections for the irregu- 
larity of the sun's motions, and thus obtain the true longitude. 

200. The corrections applied to the mean motions of a heav- 

* Biot. 



112 THE SUN. 

enly body, in order to obtain its true place, are called Equations. 
Tims the elliptical form of the earth's orbit, the precession of 
the equinoxes, and the nutation of the earth's axis, severally 
affect the place of the sun in his apparent orbit, for which 
equations are applied. In a collection of Astronomical Tables, 
a large part of the whole are devoted to this object. They give 
us the amount of the corrections to be applied under all the 
circumstances and constantly varying relations in which the 
sun, moon, and earth are situated with respect to each other. 
The angular distance, of the earth or any planet from its peri- 
helion, on the supposition that it moves uniformly in a circle, 
is called its Mean Anomaly : its actual distance at the same 
moment in its orbit is called its True Anomaly. 

Thus in figure 44, let AEP represent the orbit of the earth 
having the sun in one of the foci at S. Upon AP describe the 
circle AMP. Let E be the place of the earth in its orbit, and 
M the corresponding place in the circle ; Fj? 44 

then the angle MCP is the mean, and 
ESP the true anomaly. The difference 
between the mean and true anomaly, 
ESP — MCP, is called the Equation of 
the Center, being that correction which 
depends on the elliptical form of the or- 
bit, or on the distance of the center of 
attraction from the center of the figure, 
that is, on the eccentricity of the orbit. It is much the great- 
est of all the corrections used in finding the sun's true longi- 
tnde, amounting, at its maximum, to nearly two degrees (1° 
55' 26".8). 

Considering the mean and true anomaly as agreeing at P, 
the true place of the earth E is in advance of the mean place 
M, because the velocity near the perihelion is greater than the 
mean velocity. This difference between the mean and true 
places (equation of the center), increases till the earth has ad- 
vanced about 90° to D, when the velocity has reached its mean 
value ; from D to A, the true place moves slower than the mean, 
until at A the gain and loss are balanced, and the true and 
mean again coincide. But, as the earth's motion is now slower 
than the average, the true place falls behind the mean, and the 
equation is negative, till the earth returns to the perihelion. 




CHAPTEE V. 

-PHASES OF THE MOON — HER 
REVOLUTIONS. 

201. Next to the Sun, the Moon naturally claims our atten 
tion. 

The Moon is an attendant or satellite to the earth, around 
which she revolves at the distance of nearly 240,000 miles. 
Her mean horizontal parallax being 57' 09", f consequently, 
sin 57' 09" : semi-diameter of the earth (3956.2) : : rad : 238,545. 
(Art. 87.) 

The moon's apparent diameter is 31/ 7", and her real diameter 
2160 miles. For, 

Ead : 238,545 :: sin 15' 33^" : 1079.6. =moon's semi-diam- 
eter. (See Fig. 26, p. 71.) 

And, since spheres are as the cubes of the diameters, the 
volume of the moon is -^ that of the earth. Her density is 
nearly f (.615) the density of the earth, and her mass (=^§ x 
.615) is about gV 

202. The moon shines by reflected light borrowed from the 
sun, and, when full, exhibits a disk of silvery brightness, di- 
versified by extensive portions partially shaded. By the aid 
of the telescope, we see undoubted signs of a varied surface, 
composed of extensive tracts of level country, and numerous 
mountains and valleys. 

203. The line which separates the enlightened from the 
dark portions of the moon's disk, is called the Terminator. 
(See Fig. 2, Frontispiece.) As the terminator traverses the 
disk from new to full moon, it appears through the telescope 
exceedingly broken in some parts, but smooth in others, indi- 
cating that some portions of the lunar surface are uneven while 

* Selenography is a word more appropriate to a description of the moon, but is 
not perhaps sufficiently familiarized by use. 
\ Baily's Astronomical Tables. 

8 



114 THE MOON. 

others are level. The broken regions appear brighter than the 
smooth tracts. The latter were formerly taken for seas, and 
received names accordingly; as A (Fig. 2, Fr.), mare humorum ; 
B, mare nubium, &c. But improved telescopes have shown 
that they are extensive plains, having inequalities which, 
though comparatively low, are permanent. That there are 
mountains, is known by the following indications. As the ter- 
minator advances over the disk, the light strikes the highest 
peaks, which appear as bright points a little way upon the dark 
part of the moon. After the terminator has passed over them, 
they project shadows away from the sun, corresponding to the 
apparent shape of the mountain, and growing shorter, as the 
rays fall more nearly vertical. And again, in the waning of 
the moon, the shadows are cast in the opposite direction, length- 
ening until the dark part of the disk reaches them, and the 
summits once more become isolated bright points, and then dis- 
appear. A view with a good telescope, continued for fifteen 
minutes, will often show perceptible changes in the position of 
the shadows, and the shape of illuminated peaks. 

204. The valleys very generally have a circular form, vary- 
ing in diameter from a mile or two up to sixty miles; the 
mountain ridge, which surrounds them, being of a ring form, 
generally much more precipitous on the inner side than the 
outer. The heights surrounding these valleys are called ring- 
mountains. One or more conical mountains frequently occu- 
py the center of the area inclosed within the ring. Some of 
the principal are, !N~o. 1. Tycho ; 2. Kepler / 3. Copernicus/ 
4. Aristarchus, &c. (Fig. 1, Fr.) There are also larger and 
less regular areas, surrounded by mountains, called bulwark 
plains. The diameter of some is 130 miles ; and they gener- 
ally have small mountains, both of the conical and ring form, 
scattered over the plain. The largest are Clavius, Walther, 
Regiomontanus, Parbueh, Alphonse, Ptolemosus. Besides the 
peculiar forms now mentioned, there are chains and spurs of 
mountains, resembling terrestrial ranges.* 



« The lunar map of Beer and Madler, 2£ feet in diameter, contains a very per- 
fect delineation of the mountains and valleys of the moon, accompanied by their 
names. 



LUNAR GEOGRAPHY. 115 

205. The appearances described in the foregoing articles 
are obviously due to differences of elevation, since they are re- 
vealed by the sun-light falling on them at considerable ob- 
liquity. They come into view in succession, as the sun rises 
upon them, during the first two quarters, or sets and leaves 
them in shade in the last two. At the full moon, the sun's 
rays and our line of vision coincide in direction, and no shad- 
ows appear, and the features of mountain and valley are only 
imperfectly seen. But a peculiarity of another kind presents 
itself at the time of full moon. There are luminous stripes ex- 
tending from several of the ring-mountains in straight lines to 
the distance of hundreds of miles. They are not ridges, since 
they cast no shadows as the terminator passes them ; and the 
difference of illumination must result from difference of reflect- 
ive power. But perhaps no satisfactory reason can be given 
for their remarkable arrangement. They are sometimes termed 
lava lines. The most extensive systems of this kind occur 
around Tycho, Copernicus, Kepler, and Anaxagoras. 

There is no appearance of fluidity on the lunar surface, nor 
any such condition as we might expect to result from the flow 
of a liquid. All inequalities are angular and rigid, instead of 
being softened down by the action of water. There are no 
changes which might be ascribed to the growth and decay of 
vegetation. No spot is ever concealed by a cloud, or dimmed 
by an impure atmosphere. The cold is probably too intense 
for the existence of a liquid substance, just as the earth would 
be," if it were destitute of an atmosphere, and thus exposed to 
the low temperature of the space through which it travels.* 

2G6. The method of estimating the height of lunar moun- 
tains is as follows : 

Let ABO (Fig. 45) be the illuminated hemisphere of the 
moon, SO a solar ray touching the moon in O, a point in the 
circle which separates the enlightened from the dark part of 
the moon. All the part ODxA will be in darkness ; but if this 
part contains a mountain MF, so elevated that its summit a[ 
reaches to the solar ray SOM, the point M will be enlightened. 

* For a fuller description of the moon's surface, seeLardner's" Hand-book on 
Meteorology and Astronomy," pp. 208-212. 



116 



THE MOON. 



Let E be the place of an observer on the earth, and suppose a 
line ES to be drawn to the sun ; then MES is the elongation 
of the point M from the sun, which is determined by observa- 
tion. The angle EMS can never differ so much as 9' from the 

Fig. 45. 




supplement of MES, on account of the great distance of the 
sun compared with EM, which is about 400 to 1. But to ob- 
tain EMS accurately, 400 : 1 : : sin MES : sin ESM, which 
angle subtracted from the supplement of MES, leaves EMS. 
Let MEO, the visual angle between the mountain top and the 
terminator, be measured by a micrometer. Then, as the dis- 
tance from the earth to the moon is known, the isosceles trian- 
gle NEO gives the length of ON. As E is very small, the 
angles at N may be considered right angles ; hence, the right- 
angled triangle OMN, in which ON and M are known, gives 
OM. Finally, from the known sides- OC and OM, in the right- 
angled triangle OMC, CM is obtained ; and OC being sub- 
tracted from it, we find MF, the height of the mountain. 



207. As OM is very small compared with OC, FM is both 
more easily and more accurately found by taking the third 
proportional to 20 C and OM. For, suppose a perpendicular 
drawn from F to OC ; then OM may be called equal to FO, 
and FM to the versed sine; hence 20C : MO : : MO : FM. 
The whole work may be finished by logarithms, without taking 



LUNAR GEOGRAPHY. U 7 

from the tables any natural number, except FM the quantity 
sought. The heights of some mountains, determined in this 
way, have been found between three and four miles. 

When the moon is exactly at quadrature, EMS is a right 
angle, and OM (being coincident with ON) is obtained direct- 
ly from the micrometrical measurement of the angle E ; from 
which FM is derived as before. 

208. Schroeter, a German astronomer, estimated the heights 
of the lunar mountains by observations on their shadows. He 
made them in some cases as high as -^tt °f * ne semi-diameter 
of the moon, that is, about 5 miles. The same astronomer 
also estimates the depths of some of the lunar valleys at more 
than four miles. Hence it is inferred that the moon's surface 
is more broken and irregular than that of the earth, its moun- 
tains being higher and its valleys deeper in proportion to the 
size of the moon than those of the earth. 

209. It has sometimes been supposed that there are slight 
indications of an atmosphere about the moon. This is proba- 
bly an error. The severest test of a perceptible atmosphere 
would be the effect on a star, at the beginning and end of its 
occultation by the moon. The star would appear to be detain- 
ed a little in its diurnal motion, just before disappearing, and 
just after reappearing, in consequence of the bending of the 
rays which come from it, as they pass the edge of the moon's 
disk, and probably some loss of light would in that case be 
perceptible . at the same moments. But the most careful ob- 
servations have failed to show any such detention ; and as to 
loss of light, the star, on coming up to the edge of the moon 
disappears all at once, with a suddenness which is startling. 
If there is an atmosphere, it cannot have a thousandth part of 
the density of the earth's.* 

210. The improbability of our ever identifying artificial 
structures' in the moon may be inferred from the fact that a 
line one mile in length in the moon subtends an angle at the 
eye of only about one second. If, therefore, works of art were 

c " Lardner. 



IIS THE MOON". 

to have a sufficient horizontal extent to become visible, they 
can hardly be supposed to attain the necessary elevation, when 
we reflect that the height of the great pyramid of Egypt is less 
than the sixth part of a mile. 

PHASES OF THE MOON. 

211. The changes of the moon, commonly called her 
Phases, arise from different portions of her illuminated side 
being turned toward the earth at different times. When the 
moon is first seen after the setting sun, her form is that of a 
bright crescent, on the side of the disk next to the sun, while 
the other portions of the disk shine with a feeble light, reflect- 
ed to the moon from the earth. Every night we observe the 
moon to be further and further eastward of the sun, and at the 
same time the crescent enlarges, until, when it has reached an 
elongation from the sun of nearly 90°, half her visible disk is 
enlightened, and she is said to be in her first quarter. The 
terminator, or line which separates the illuminated from the 
dark part of the moon, is convex toward the sun from the 
new moon to the first quarter, and the moon is said to be 
horned. The extremities of the crescent are called cusps. At 
the first quarter, the terminator becomes a straight line, coin- 
ciding with a diameter of the disk ; but after passing this 
point, the terminator becomes concave toward the sun, bound- 
ing that side of the moon by an elliptical curve, when the 
moon is said to be gibbous. When the moon arrives at the 
distance of 180° from the sun, the entire circle is illuminated, 
and the moon is full. She is then in opposition to the sun, 
rising about the time the sun sets. For a week after the full, the 
moon appears gibbous again, until, at a little less than 90° 
from the sun, she resumes the same form as at the first quarter, 
being then at her third quarter. From this time until new 
moon, she exhibits again the form of a crescent before the 
rising sun, until approaching her conjunction with the sun, 
her narrow thread of light is lost in the solar blaze; and final- 
ly, at the moment of passing the sun, the dark side is wholly 
turned toward us and for some time we lose sight of the moon. 

The two points in the orbit corresponding to new and full 
moon respectively, are called by the common name of syzygies ; 



PHASES. 119 

those which are 90° degrees from the sun are called quadra- 
tures j and the points half way between the syzygies and quad- 
ratures are called octants. The circle which divides the en- 
lightened from the unenlightened hemisphere of the moon, is 
called the circle of illumination : that which divides the hem- 
isphere that is turned toward us from the opposite one is called 
the circle of the disk. The degree of each phase depends on 
the angle between these circles. 

212. As the moon is an opaque body of a spherical figure, 
and borrows her light from the sun, it is obvious that that 
half only which is toward the sun can be illuminated. More 
or less of this side is turned toward the earth, according as 
the moon is at a greater or less elongation from the sun. The 
reason of the different phases will be best understood from a 

Fig. 46. 




diagram. Therefore, let T (Fig. 46) represent the earth, and 
S the sun. Let A, B, C, &c, be successive positions of the 
moon. At A the entire dark side of the moon being turned 
toward the earth, the disk would be wholly invisible. At B, 
the circle of the disk cuts off a small part of the enlightened 
hemisphere, which appears in the heavens at 5, under the form 
of a crescent. At C, the first quarter, the circle of the disk 
cuts off half the enlightened hemisphere, and the moon appears 
dichotomized at c. In like manner it will be seen that the 
appearances presented at D, E, F, &c, must be those repre- 
sented at d, e,f. 



120 THE MOON. 



REVOLUTIONS OF THE MOON. 



213. The moon revolves around the earth from west to east, 
making the entire circuit of the heavens in about 27 \ days. 

The precise law of the moon's motions in her revolution 
around the earth is ascertained, as in the case of the sun (Art. 
155), by daily observations on her meridian altitude and right 
ascension. Thence are deduced by calculation her latitude 
and longitude, from which we find, that the moon describes 
on the celestial sphere a great circle of which the earth is the 
center. 

The period of the moon's revolution from any point in the 
heavens round to the same point again, is called a month. A 
sidereal month is the time of the moon's passing from any star, 
until it returns to the same star again. A sy nodical month* is 
the time from one conjunction or new moon to another. The 
synodical month is about 29} days, or more exactly, 29d. 12h. 
44m. 2 S .8= 29.53 days. The sidereal month is about two days 
shorter, being 27d. 7h. 43m. ll s .5= 27.32 days. As the sun 
and moon are both revolving in the same direction, and the 
sun is moving nearly a degree a day, during the 27 days of the 
moon's revolution, the sun must have moved 27°. Now since 
the moon passes over 360° in 27.32 days, her daily motion 
must be 13° 17'. It must therefore evidently take about two 
days for the moon to overtake the sun. The difference between 
these two periods may, however, be determined with great 
exactness. The middle of an eclipse of the sun marks the 
time of conjunction or new moon ; and by dividing the inter- 
val between any two distant solar eclipses by the number of 
revolutions of the moon, or lunations, we obtain the precise 
period of the synodical month. Suppose, for example, two 
eclipses occur at an interval of 1,000 lunations ; then the whole 
number of days and parts of a day that compose the interval 
divided by 1,000 will give the exact time of one lunation. f 
The time of the synodical month being ascertained, the exact 
period of the sidereal month may be derived from it. For the 

* aw and o<5os, implying that the two bodies come together. 

f It might at first view seem necessary to know the period of one lunation be- 
fore we could know the number of lunations in any given interval. This period 
is known very nearly from the interval between one new moon and another. 



REVOLUTIONS. 1 21 

arc which the moon describes in order to come into conjunc- 
tion with the sun, exceeds 360° by the space which the sun has 
passed over since the preceding conjunction, that is, in 29.53 
days. Therefore, 

365.24 : 360° : : 29.53 : 29°.l=arc which the moon must de- 
scribe more than 360° in order to overtake the sun. Hence, as 
the whole distance the moon must move from the sun to reach 
it again, is to one revolution of the moon, so is the time of ac- 
complishing the former, to the time of a revolution; i. e., 
360°+29°.l : 360° : : 29.53d. : 27.32d. 

214.. The moon's orbit is inclined to the ecliptic at an angle 
of about 5° (5° 8' 48"). The intersections are called the ascend- 
ing and descending nodes : through the ascending, the moon 
passes from south to north ; through the descending, from 
north to south. The angle is found by measuring the moon's 
greatest latitude, which is, of course, equal to the inclination 
of the circles. 

215. The moon, at the same age, crosses the meridian at 
different altitudes at different seasons of the year. The full 
moon, for example, will appear much further in the south 
when on the meridian at one period of the year than at anoth- 
er. This is owing to the fact that the moon's path is different- 
ly situated with respect to the horizon, at a given time of 
night, at different seasons of the year. By taking the ecliptic 
on an. artificial globe to represent the moon's path (which is 
always near it, Art. 214), and recollecting that the new moon 
is seen in the same part of the heavens with the sun, and the 
full moon in the opposite part of the heavens from the sun, we 
shall readily see that in the winter the new moons must run 
low because the sun does, and for a similar reason the full 
moons mast run high. It is equally apparent that, in summer, 
when the sun runs high, the new moons must cross the merid- 
ian at a high, and the full moons at a low altitude. This 
arrangement gives us a great advantage in respect to the 
amount of light received from the moon ; since the full moon 
is longest above the horizon during the long nights of winter. 
when her presence is most needed. This circumstance is es- 
pecially favorable to the inhabitants of the polar regions, the 



122 THE MOON. 

moon, when full, traversing that part of her orbit which lies 
north of the equator, and of course above the horizon of the 
north pole, and traversing the portion that lies south of the 
equator, and below the polar horizon, when new. During the 
polar winter, therefore, the moon, from the first to the last 
quarter, is commonly above the horizon, while the sun is ab- 
sent ; whereas, during summer, while the sun is present, the 
moon is above the horizon while describing her first and last 
quadrants. 

216. About the time of the autumnal equinox, the moon 
when near the full, rises about sunset for a number of nights 
in succession; and as this is, in England, the period of har- 
vest, the phenomenon is called the Harvest Moon. To un- 
derstand the reason of this, since the moon is never far from 
the ecliptic, we will suppose her progress to be in the ecliptic. 
If the moon moved in the equator, then, since this great circle 
is at right angles to the axis of the earth, all parts of it, as the 
earth revolves, would cut the horizon at the same constant angle. 
But the moon's orbit, or the ecliptic, which is here taken to 
represent it, being oblique to the equator, cuts the horizon at 
different angles in different parts, as will easily be seen by 
reference to an artificial globe. When the first of Aries, or 
vernal equinox, is in the eastern horizon, it will be seen that 
the ecliptic (and consequently the moon's orbit) makes its 
least angle with the horizon. Now at the autumnal equinox, 
the sun being in Libra, the moon at the full is in Aries, and 
rises when the sun sets. On the following evening, although 
she has advanced in her orbit about 13° (Art. 213), yet her 
progress being oblique to the horizon, and at a small angle 
with it, she will be found at this time but a little way below 
the horizon, compared with the point where she was at sunset 
the preceding evening. She therefore rises but a little later 
each evening than she did on the evening previous, but her 
place of rising moves rapidly northward. It should be ob- 
served, that in making her revolution round the earth, the 
moon must pass the first of Aries, and therefore make these 
small differences in the time of rising, every month. But as 
the moon is not full at the same time, except in autumn, the 
circumstance attracts no attention. 




REVOLUTIONS. 123 

"When the ascending node is at the vernal equinox, the 
angle of the moon's path with the horizon at the time of rising, 
is 10° smaller than when the descending node is there ; and the 
intervals vary accordingly. This occurs once in about 18 years. 

217. The moon is about }q nearer to us when near the zenith 
than when in the horizon. 

The horizontal distance CD (Fig 47) is Fig. 47. 

nearly equal to AD=AD', which is great- 
er than CD' by AC, the semi-diameter of 
the earth =6 J o the distance of the moon. 
Accordingly, the apparent diameter of the 
moon, when actually measured, is about D 
30" (which equals about -£$ of the whole) greater when in the 
zenith than in the horizon. The apparent enlargement of the 
full moon when rising, is owing to the same causes as that of 
the sun, as explained in article 96. 

218. The moon turns on its axis in the same time in which 
it revolves around the earth. 

This is knowm by the moon's always keeping nearly the 
same face toward us, as is indicated by the telescope, which 
could not happen unless her revolution on her axis kept pace 
with her motion in her orbit. Thus, it will be seen by in- 
specting figure 31, that the earth turns different faces toward 
the sun at different times ; and if a ball having one hemi- 
sphere white and the other black be carried around a lamp, it 
will easily be seen that it cannot present the same face con- 
stantly toward the lamp unless it turns once on its axis while 
performing its revolution. But though the same side of the 
moon on the whole is always toward the earth, yet there are 
small apparent oscillations, by which narrow portions of the 
remote side are presented alternately to view. These are 
called Vibrations. 

219. One is the libralion in longitude, because it brings 
into view portions of the equator on one side and then on the 
other. It is owing to the fact that the moon revolves uni- 
formly on its axis, and with unequal angular motion around 
the earth. Near the apogee, where she advances slowest, she 



12± THE MOON. 

describes less than 90° of her orbit while she turns just one- 
fourth round upon her axis ; consequently showing us a little 
of the further side on the east limb. But in the perigeal part 
of her orbit, she advances faster than the mean, and therefore 
in one-fourth of her diurnal rotation she moves forward more 
than 90° in her orbit, and presents some surface beyond the 
west limb. 

The libration 'in latitude, by which she alternately presents 
to our view the space about her poles, is caused by the obliqui- 
ty of her equator to her orbit. Her equator is inclined about 
1^° (1° 30' il") to the ecliptic, and remains parallel to itself. 
But the angle between her equator and orbit varies from about 
3^-° to 6-J°, on account of the morion of the nodes. Each pole 
of the moon is presented toward the earth every 27 days, just 
as the earth's poles are turned toward the sun every year, 
though in a much less degree. (See Fig. 31.) 

The moon exhibits another phenomenon of this kind, called 
her diurnal libration, depending on the daily rotation of the 
spectator. She turns the same face toward the center of the 
earth only, whereas we view her from the surface. When she 
is on the meridian, we see her disk nearly as though we 
viewed it from the center of the earth, and hence in this situa- 
tion it is subject to little change ; but when near the horizon, 
our circle of vision takes in more of the upper limb than 
would be presented to a spectator at the center of the earth. 
Hence, from this cause, we see a portion of one limb while the 
moon is rising, which is gradually lost sight of, and we see a 
portion of the opposite limb as the moon declines toward the 
west. It will be remarked that neither of the foregoing 
changes implies any actual motion in the moon, but that each 
arises from a change of position in the spectator relative to the 
moon. 

220. An inhabitant of the moon would have but one day 
and one night during the whole lunar month of 29|- days. 
One of its days, therefore, is equal to nearly 30 of ours. So 
protracted an exposure to the sun's rays, if the moon had an 
atmosphere like that of the earth, would occasion an excessive 
accumulation of heat ; and so long an absence of the sun must 
occasion a corresponding degree of cold. Each day would be 



REVOLUTION-. 125 

a wearisome summer ; each night a severe winter.* A spec- 
tator on the side of the moon which is opposite to us would 
never see the earth ; but one on the side next to us would see 
the earth presenting a gradual succession of changes during 
his long night of 360 hours. Soon after the earth's conjunc- 
tion with the sun, he would have the light of the earth re- 
flected to him, presenting at first a crescent, but enlarging, as 
the earth approaches its opposition, to a great orb, 13 times as 
large as the full moon appears to us, and affording nearly 13 
times as much light. Our seas, plains, mountains, and clouds, 
would present a great diversity of appearance, as the earth 
performed its diurnal rotation ; though the distinctness would 
be much impaired by the strong light reflected by our atmos- 
phere. The earth to his view would remain always in the 
same part of the sky, having only small monthly oscillations, 
north and south, by means of the libration in latitude, also 
east and west, by the libration in longitude. For, being un- 
conscious of his own motion around the earth, it would seem 
to revolve about his planet from west to east; but, meanwhile, 
his own diurnal rotation would give the earth an apparent 
motion to the west at the same mean rate, and the two would 
balance each other, except so far as the librations would affect 
them. An observer on the center of the moon's disk, would 
see the earth always over head ; one at the edge of the disk, 
would see it at the horizon. The earth is full to the moon 
when the latter is new to us ; and universally the two phases 
are complementary to each other. 

221. If the ecliptic (the earth's path about the sun), and 
the moon's path about the earth, were visible lines in the sky, 
since we are in the plane of each, they would both appear as 
great circles intersecting each other in opposite points, and in- 
clined about 5° to each other. But if we could take a view of 
these orbits from a distant position out of their planes, they 
would appear as two very unequal circles, one having a diam- 
eter 400 times greater than the other, and the small circle 
moving around with its center upon the circumference of the 
large one, once in a year. 

* Francoeur, Uranog., p. 91. 



126 THE MOON, 

But again, if we confine our attention to the moon's patli in 
its relations to the sun, instead of the earth, and trace it as a 
visible line in the solar system, we shall see that it loses en- 
tirely its character of a small circle on the circumference of a 
large one. On the other hand, it can hardly be distinguished 
from the earth's orbit itself, making 25 slight undulations al- 
ternately inside and outside of it, and never deviating from it 
more than one 400th part of its radius, as represented in 
Fig. 4:7'. 

Fig. 47*. 



If we regard now the forces to which the moon is subjected 
in describing this path, it is clear, that since it differs so little 
from that of the earth, the moon must be controlled by nearly 
the same forces as those which keep the earth in its orbit ; so 
that, if the earth were annihilated, the moon would still pre- 
serve its course round the sun, with so little change from its 
present orbit, that it would hardly be noticed by an observer 
who could take the whole into view at once. According to 
the second law of motion (Nat. Phil., Art. 12), the moon's mo- 
tion round the earth simply co-exists with its motion round the 
sun ; and the joint effect is a curve of that species called an 
epicycloid. 

,222. The relative attraction exerted by the sun and earth 
upon the moon, is found by applying the formula (Art. 177) 

fco . Calling the radius of the moon's orbit=l, that of the 

earth's is about 400 ; and the times are 27.32 and 365.25. 

Hence, attraction to the sun : that to the earth : : /ft _^ -.„. 5 : 
' (36o.2o)" 

— : : 2.2 : 1, So that the sun, though so very far from 



(27.32) 

the moon, exerts upon it 2J times more attraction than the 
earth does. The moon, therefore, is much more under the in- 
fluence of the sun than of the earth ; the latter only causing it 



LUNAR IRREGULARITIES. 127 

to oscillate on her own path as already shown. When the 
moon is in conjunction, the attraction of the earth diminishes 
its tendency to the sun, and therefore its distance increases, till 
it comes to the opposition ; and while thus ascending, being 
in the rear of the earth, it is accelerated by it, and goes past it 
at the moment of opposition. But now the attraction of the 
earth conspires with that of the sun, so that the moon can not 
keep in a circular orbit at that distance, and therefore descends 
again toward the conjunction. While descending, being now 
in advance of the earth, it is retarded by it, and loses so much 
of its velocity, that on reaching the conjunction, the earth 
passes it again as before. Thus the moon, by the earth's 
influence, approaches the sun, and then recedes from it, and 
also gains velocity, and then loses it, suffering these changes 
as slight disturbances in its great annual revolution about the 
sun. 

The earth is also affected in precisely the same manner, 
though in a far less degree*, by the moon ; it is the center of 
gravity of the two, which describes the annual elliptical orbit. 

223. If a chord be supposed drawn in the earth's orbit 
(Fig. 47'), between successive points of quadrature, it is found 
that this chord, at its middle point, falls about 600,000 miles 
within the orbit, while the moon is only 238,000 miles within 
it at the conjunction. Her path is, therefore, so near that of 
the earth as to be always concave toward the sun. 



CHAPTER VI. 

LUNAR IRREGULARITIES. 

224. Considering the moon's motion simply in its relation 
to the earth, we have thus far spoken of its path as an ellipse, 
with the earth in one focus. But careful observations have 
proved that this elliptical revolution is subject to numerous 
irregularities. The law of universal gravitation has been ap- 
plied with wonderful success to their investigation, and its 



128 THE MOON. 

results have conspired, with those of long-continued observa- 
tion, to furnish the means of ascertaining, with great exactness, 
the place of the moon in the heavens at any given instant of 
time, past or future, and thus to enable astronomers to deter- 
mine longitudes, to calculate eclipses, and to solve various other 
problems of the highest interest. A complete understanding 
of all the irregularities of the moon's motions must be sought 
for in more extensive treatises of astronomy than the present ; 
but some general acquaintance with the subject, clear and in- 
telligible, as far as it goes, may be acquired by first gaining a 
distinct idea of the mutual actions of the sun, the moon, and 
the earth. 

225. The irregularities of the moon's motions are due chiefly 
to the disturbing influence of the sun, which operates in two 
ways : first, by acting unequally on the earth and moon, and, 
secondly, by acting obliquely on the moon. 

If the sun acted equally on the earth and moon, and always 
in parallel lines, this action would serve only to restrain them 
in their annual motions round the sun, and would not affect 
their actions on each other, or their motions about their com- 
mon center of gravity. In that case, if they were allowed to 
fall directly toward the sun, they would fall equally, and their 
respective situations would not be affected by their descending 
equally toward it. We might then conceive them as in a 
plane, every part of which being equally acted on by the sun, 
the whole plane would descend toward the sun, but the re- 
spective motions of the earth and the moon in this plane w T ould 
be .the same as if it were quiescent. Supposing, then, this 
plane and all in it to have an annual motion imprinted on it, 
it would move regularly round the sun, while the earth and 
moon would move in it, with respect to each other, as if the 
plane were at rest, without any irregularities. But because 
the moon is nearer the sun in one half of her orbit than the 
earth is, and in the other half of her orbit is at a greater dis- 
tance than the earth from the sun, while the power of gravity 
is always greater at a less distance, it follows, that in one-half 
of her orbit the moon is more attracted than the earth toward 
the sun, and in the other half less attracted than the earth. 
The excess of the attraction, in the first case, and the defect in 



LUNAR IRREGULARITIES. 129 

the second, constitutes a disturbing force; to which we may 
add another, namely, that arising- from the oblique action of 
the solar force, since this action is not directed in parallel 
lines, but in lines that meet in the center of the sun ; and one 
part of this oblique action is in the orbit of the moon, being 
now east of the earth, and then west of it ; and the other part 
is from the orbit toward the plane of the ecliptic. 

* 

226. To see the effects of this process, let us suppose that 

the projectile motions of the earth and moon were destroyed, 
and that they were allowed to fall freely toward the sun. If 
the moon was in conjunction with the sun, or in that part of 
her orbit which is nearest to him, the moon would be more 
attracted than the earth, and fall with greater velocity toward 
the sun ; so that the distance of the moon from the earth would 
be increased in the fall. If the moon was in opposition, or in 
the part of her orbit which is farthest from the sun, she would 
be less attracted than the earth by the sun, and would fall 
with a less velocity toward the sun, and would be left behind ; 
so that the distance of the moon from the earth would be in- 
creased in this case also. If the moon was in one of the quar- 
ters, then the earth and moon being both attracted toward the 
center of the sun, they would both descend directly toward 
that center, and by approaching it, they would necessarily, at 
the same time, approach each other, and in this case their 
distance from each other would be diminished. Now when- 
ever the action of the sun would increase their distance, if 
they were allowed to fall toward the sun, then the sun's action, 
by endeavoring to separate them, diminishes their gravity to 
each other ; whenever the sun's action would diminish the 
distance, then it increases their mutual gravitation. Hence, 
in the conjunction and opposition, that is, in the syzygies, 
their gravity toward each other is diminished by the action of 
the sun, while in the quadratures it is increased. But it must 
be remembered that it is not the total action of the sun on 
them that disturbs their motions, but only that part of it 
which tends at one time to separate them, and at another time 
to bring them nearer together. The other and far greater part, 
has no other effect than to retain them in their annual course 
around the sun. 

9 



130 



THE MOON. 



227. Suppose the moon setting out from the quarter that 
precedes the conjunction with a velocity that would make her 
describe an exact circle round the earth, if the sun's action had 
no effect on her : since her gravity is increased by that action, 
she must descend toward the earth and move within that cir- 
cle. Her orbit then would be more curved than it otherwise 
would have been ; because the addition to her gravity will 
make her fall further at the end of an arc below the tangent 
drawn at the other end of it. Her motion will be thus accel- 
erated, and it will continue to be accelerated until she arrives 
at the ensuing conjunction, because the direction of the sun's 
action upon her, during that time, makes an acute angle with 
the direction of her motion. (See Fig. 41.) At the conjunc- 
tion, her gravity toward the earth being diminished by the 
action of the sun, her orbit will then be less curved, and she 
will be carried further from the earth as she moves to the next 
quarter; and because the action of the sun makes there an 
obtuse angle with the direction .of 

her motion, she will be retarded in Flg - 48 - 

the same degree as she was acceler- 
ated before. 

228. After this general explana- 
tion of the mode in which the sun 
acts as a disturbing force on the mo- 
tions of the moon, the learner will 
be prepared to understand the math- 
ematical development of the same 
doctrine. 

Therefore, let ADBC (Fig. 48) be 
the orbit, nearly circular, in which 
the moon M revolves in the direc- 
tion CADB, round the earth E. 
Let S be the sun, and let SE, the 
radius of the earth's orbit, be taken 
to represent the force with which 
the earth gravitates to the sun. 

Then (Art. 181) ~™ : ^-^ : : SE : ^^ = the force by which 

the sun draws the moon in the direction MS. Take MG = 




LUNAR IRREGULARITIES. 131 

SE 3 

Q^ 23 and let the parallelogram KF be described, having MG 

for its diagonal* and having its sides parallel to EM and ES. 
The force MG may be resolved into two, MF and MK, of 
which MF, directed toward E, the center of the earth, in- 
creases the gravity of the moon to the earth, and does not 
hinder the areas described by the radius vector from being 
proportional to the times. The other force MK draws the 
moon in the direction of the line joining the centers of the 
sun and earth. It is, however, only the excess of this force 
above the force represented by SE, or that which draws the 
earth to the sun, which disturbs the relative position of the 
moon and earth. This is evident, for if KM were just equal to 
ES, no disturbance of the moon, relative to the earth, could 
arise from it. If, then, ES be taken from MK, the difference 
HK is the whole force in the direction parallel to SE, by which 
the sun disturbs the relative position of the moon and earth. 
Now, if in MK, MN be taken equal to HK, and if NO be 
drawn perpendicular to the radius vector EM produced, the 
force MIST may be resolved into two, MO and ON, the first 
lessening the gravity of the moon to the earth ; and the second, 
being parallel to the tangent of the moon's orbit in M, accel- 
erates the moon's motion from C to A, and retards it from A 
to D, and so alternately in the other two quadrants. Thus the 
whole solar force directed to the center of the earth, is com- 
posed of the two parts MF and MO, which are sometimes op- 
posed to "one -another, but which never affect the uniform 
description of the areas about E. Near the quadratures the 
force MO vanishes, and the force MF, which increases the 
gravity of the moon to the earth, coincides with CE or DE. 
As the moon approaches the conjunction at A, the force MO 
prevails over MF, and lessens the gravity of the moon to the 
earth. In the opposite point of the orbit, when the moon is in 
opposition at B, the force with which the sun draws the moon 
is less than that with which the sun draws the earth, so that 
the effect of the solar force is to separate the moon and earth, 
or to increase their distance ; that is, it is the same as if, con- 
ceiving the earth not to be acted on, the sun's force drew the 
moon in the direction from E to B. This force is negative, 
therefore, in respect to the force at A, and the effect in both 



132 THE MOON. 

cases is to draw the moon from the earth in a direction perpen- 
dicular to the line of the quadratures. Hence, the general 
result is, that by the disturbing force of the sun, the gravity to 
the earth is increased at the quadratures, and diminished at 
the syzygies. It is found by calculation that the average 
amount of this disturbing force is 5^ of the moon's gravity to 
the earth.* 

229. With, these general principles in view, we may now 
proceed to investigate the figure of the moon's orbit, and the 
irregularities to which the motions of this body are subject. 

230. The figure of the moon's orbit is an ellipse r having the 
earth in one of the foci. 

The elliptical figure of the moon's orbit, is revealed to us by 
observations on her changes in apparent diameter, and in her 
horizontal parallax. First, we may measure from day to day 
the apparent diameter of the moon. Its variations being in- 
versely as the distances (Art. 163), they give us at once the 
relative distance of each point of observation from the focus. 
Secondly, the variations on the moon's horizontal parallax, 
which also are inversely as the distances (Art. 82),, lead to the 
same results. Observations on the angular velocities, com- 
bined with the changes in the lengths of the radius vector, af- 
ford the means of laying down a plot of the lunar orbit, as in 
the case of the sun, represented in figure 32. The orbit is 
shown to be nearly an ellipse, because it is found to have the 
properties of an ellipse. 

The moon's greatest and least apparent diameters are respect- 
ively 33'. 518 and 29'. 365, while her corresponding changes of 
parallax are 61 '.4 and 53'. 8. The two ratios ought to be equal, 
and we shall find such to be the fact very nearly, as expressed 
by the foregoing numbers ; for, 

61.4: 53.8 :: 33*518 : 29.369. 

The greatest and least distances of the moon from the earth, 
derived from the parallaxes, are 63.8419 and 55.9164, or nearly 
64 and 56, the radius of the earth being taken for unity. 
Hence, taking the arithmetical mean, which is 59.879, we find 

* Play fair. 



LUNAR IRREGULARITIES. 133 

that the mean distance of the moon from the earth is very 
nearly 60 times the radius of the earth. The point in the 
moon's orbit nearest the earth, is called her perigee • the point 
furthest from the earth, her apogee. 

The greatest and least apparent diameters of the sun are re- 
spectively 32.583, and 31.517, which numbers express also the 
ratio of the greatest and least distances of the earth from the 
sun. By comparing this ratio with that of the distances of the 
moon, it will be seen that the latter vary much more than the 
former, and consequently that the lunar orbit is much more ec- 
centric than the solar. The eccentricity of the moon's orbit is 
in fact 0.0548 (the semi-major axis being as usual taken for 
unity)= T 1 ¥ of its mean distance from the earth, while that of 
the earth is only .01685 =%■§ of its mean distance from the sum 

231. The moorHs nodes constantly shift their positions in 
the eeliptic from east to west, at the rate of 19° S^ per annum^ 
returning io the same points in 18.6 years. 

Suppose the great circle of the ecliptic marked out on the 
face of the sky in a distinct line, and let us observe, at any 
given time, the exact point where the moon crosses this line, 
which we will suppose to be close to a certain star ; then, on 
its next return to that part of the heavens, we shall find that 
it crosses the ecliptic sensibly to the westward of that star, and 
so on, further and further to the westward every time it crosses 
the ecliptic at either node. This fact is expressed by saying 
that the nodes retrograde on the ecliptic, and that the line which 
joins them, or" the line of the nodes, revolves from east to west. 

232. This shifting of the moon's nodes implies that the lu- 
nar orbit is not a curve returning into itself, but that it more 
resembles a spiral like the curve represented in figure 49, where 
abc represents the ecliptic, and ABC Flg> 49> 

the lunar orbit, having its nodes at b 

C and E, instead of A and a\ con- /^^^^^^s. 

sequently, the nodes shift backward v", Y\ 

through the arcs aG and AE. The \ ( J) e 

manner in which this effect is pro- V W.. ^^' 

duced may be thus explained. That ^f^zrr^ 

part oi the solar force which is parallel to the line joining the 



134: THE MOON. 

centers of the sun and earth (see Fig. 4S), is not in the plane 
of the moon's orbit (since this is inclined to the ecliptic about 
5°), except when the sun itself is in that plane, or when the 
line of the nodes being produced passes through the sun. In 
all other cases it is oblique to the plane of the orbit, and may 
be resolved into two forces, one of which is at right angles to 
that plane, and is directed toward the ecliptic. This force of 
course draws the moon continually toward the ecliptic, or pro- 
duces a continual deflection of the moon from the plane of her 
own orbit toward that of the earth. Hence the moon meets 
the plane of the ecliptic sooner than it would have done if that 
force had not acted. At every half revolution, therefore, the 
point in which the moon meets the ecliptic shifts in a direction 
contrary to that of the moon's motion, or contrary to the order 
of the signs. If the earth and sun were at rest, the effect 
of the deflecting force just described would be to produce a 
retrograde motion of the line of the nodes till that line was 
brought to pass through the sun, and, of consequence, the plane 
of the moon's orbit to do the same, after which they would 
both remain in their position, there being no longer any force 
tending to produce change in either. But the motion of the 
earth carries the line of the nodes out of this position, and pro- 
duces, by that means, its continual retrogradation. The same 
force produces a small variation in the inclination of the moon's 
orbit, giving it an alternate increase and decrease within very 
narrow limits.* These points will be easily understood by the 
aid of a diagram. . Therefore, let MN (Fig. 50) be the ecliptic, 
AE"B the orbit of the moon, the moon being in L, and N its 
descending node. Let the disturbing force of the sun which 
tends to bring it down to the ecliptic be represented by L5, and 
its velocity in its orbit by ~La. Under the action of these two 
forces, the moon will describe the diagonal Lc of the parallelo- 
gram ba, and its orbit will be changed from AN to ~LN' ; the 
node N passes to N' ; and the exterior angle at W of the tri- 
angle LNJSP being greater than the interior and opposite angle 
at N, the inclination of the orbit is increased at the node. 
After the moon has passed the ecliptic to the south side to I, 
the disturbing force Id produces a new change of the orbit "N'le 

* Playfair. 



LUNAR IRREGULARITIES. 

Fig. 50. 



135 




to Wlf, and the inclination is diminished as at W r . In general, 
while the moon is receding from one of the nodes, its inclina- 
tion is diminishing ; while it is approaching a node, the incli- 
nation is increasing.* 

233. The period occupied by the sun in passing from one 
of the moon's nodes until it comes round to the same node 
again, is called the synodical revolution of the node. This 
period is shorter than the sidereal year, being only about 346^- 
days. For since the node shifts its place to the westward 19° 
35' per annum, the sun, in his annual revolution, comes to it 
so much before he completes his entire circuit ; and since the 
sun moves about a degree a day, the synodical revolution of 
the node is 365—19 = 346, or more exactly, 346.619851. The 
time from one new moon, or from one full moon, to another, 
is 29.5305887 days. Now 19 synodical revolutions of the nodes 
contain very nearly 223 of these periods. 

For 346.619851 xl9=6585.78, 
And 29.5305887x223=6585.32. 

Hence, if the sun and moon were to leave the moon's node 
together, after the sun had been round to the same node 19 
times, the moon would have performed very nearly 223 synod- 
ical revolutions, and would, therefore, at the end of this period 
meet at the same node, to repeat the same circuit. And since 
eclipses of the sun and moon depend upon the relative position 
of the sun, the moon, and node, these phenomena are repeated 
in nearly the same order, in each of those periods. Hence, 



* Francceur, Uianog., p. 158 ; Robison's Phys. Astronomy. Art. 204. 



136 THE MOON. 

this period, consisting of about 18 years and 10 days, under 
the name of the Saros, was used by the Chaldeans and other 
ancient nations in predicting eclipses. 

234. The Metonio Cycle is not the same with the Saros, 
but consists of 19 tropical years. During this period the moon 
makes very nearly 235 synodical revolutions, and hence the 
new and full moons , if reckoned by periods of 19 years, recur 
at the same dates. If, for example, a new moon fell on the 
fiftieth day of one cycle, it would also fall on the fiftieth day 
of each succeeding cycle; and, since the regulation of games, 
feasts, and fasts, has been made very extensively according to 
new or full moons, hence this lunar cycle has been much used 
both in ancient and modern times. The Athenians adopted it, 
433 years before the Christian era, for the regulation of their 
calendar, and had it inscribed in letters of gold on the walls 
of the temple of Minerva. Hence the term Golden Number, 
which denotes the year of the lunar cycle. 

235. The line of the apsides of the moon's orbit revolves 
from west to east through her whole orbit in about nine years. 

If, in any revolution of the moon, we should accurately mark 
the place in the heavens where the moon comes to its perigee 
(Art. 230), we should find that at the next revolution it would 
come to its perigee at a point a little further eastward than 
before, and so on at every revolution, until, after 9 years, it 
would come to its perigee at nearly the same point as at first. 
This fact is expressed by saying that the perigee, and of course 
the apogee, revolves, and that the line which joins these two 
points, or the line of the apsides, also revolves. 

The place of the perigee may be found by observing when 
the moon has the greatest apparent diameter. But as the 
magnitude of the moon varies slowly at this point, a better 
method of ascertaining the position of the apsides, is to take 
two points in the orbit where the variations in apparent diam- 
eter are most rapid, and to find where they are equal on oppo- 
site sides of the orbit. The middle point between the two will 
give the place of the perigee. 

The angular distance of the moon from her perigee in any 
part of her revolution, is called the Maoris Anomaly, 



LUNAR IRREGULARITIES. 137 

236. The change of place in the apsides of the moon's 
orbit, like the shifting of the nodes, is caused by the disturbing 
influence of the sun. If when the moon sets out from its peri- 
gee, it were urged by no other force than that of projection, 
combined with its gravitation toward the earth, it would de- 
scribe a symmetrical curve (Art. 187), coming to its apogee at 
the distance of 180° But as the mean disturbing force in the 
direction of the radius vector tends, on the whole, to diminish 
the gravitation of the moon to the earth, the portion of the 
path described in an instant will be less deflected from her 
tangent, or less curved, than if this force did not exist. Hence 
the path of the moon will not intersect the radius vector at 
right angles, that is, she will not arrive at her apogee until after 
passing more than 180° from her perigee, by which means 
these points will constantly shift their positions from west to 
east.* The motion of the apsides is found to be 3° 1' 20" for 
every sidereal revolution of the moon. 

237. On account of the greater eccentricity of the moon's 
orbit above that of the sun, the Equation of the Center, or that 
correction which is applied to the moon's mean anomaly to find 
her true anomaly (Art. 200), is much greater than that of the 
sun, being when greatest more than six degrees (6° 17' 12".7), 
while that of the sun is less than two degrees (1° 55' 2 6". 8). 

The irregularities in the motions of the moon may be com- 
pared to. those of the magnetic needle. As a first approxima- 
tion, we say that the needle places itself in a north and south 
line*. On closer examination, however, we find that, at differ- 
ent places, it deviates more or less from this line, and we in- 
troduce the first great correction under the name of the decli- 
nation of the needle. But observation shows us that the 
declination alternately increases and diminishes every day, and 
therefore we apply to the declination itself a second correction 
for the diurnal variation. Finally, we ascertain, from long- 
continued observations, that the diurnal variation is affected 
by the change of seasons, being greater in summer than in 
winter, and hence we apply to the diurnal variation a third 
correction for the annual variation. 

* Playfair. 



138 THE MOON. 

In like manner, we shall find the greater inequalities of the 
moon's motions are themselves subject to subordinate inequal- 
ities which give rise to smaller equations, and these to smaller 
still, to the last degree of refinement. 

238. Next to the equation of the center, the greatest cor- 
rection to be applied to the moon's longitude, is that which 
belongs to the Ejection. The evection is a change of form in 
the lunar orbit, by which its eccentricity is sometimes increased, 
and sometimes diminished. It depends on the position of the 
line of the apsides with respect to the line of the syzygies. 

This irregularity, and its connection with the place of the 
perigee with respect to the place of conjunction or opposition, 
was known as a fact to the ancient astronomers, Hipparchus 
and Ptolemy ; but its cause was first explained by Newton in 
conformity with the law of universal gravitation. It was 
found, by observation, that the equation of the center itself 
was subject to a periodical variation, being greater than its 
mean, and greatest of all when the conjunction or opposition 
takes place at the perigee or apogee, and least of all when the 
conjunction or opposition takes place at a point half way be- 
tween the perigee and apogee ; or, in the more common lan- 
guage of astronomers, the equation of the center is increased 
when the line of the apsides is in syzygy, and diminished when 
that line is in quadrature. If, for example, when the line of 
the apsides is in syzygy, we compute the moon's place by de- 
ducting the equation of the center from the mean anomaly (see 
Art. 200), seven days after conjunction, the computed longi- 
tude will be greater than that determined by actual observa- 
tion, by about 80 minutes ; but if the same estimate is made 
when the line of the apsides is in quadrature, the computed 
longitude will be less than the observed, by the same quantity. 
These results plainly show a connection between the velocity 
of the moon's motions and the position of the line of the apsides 
with respect to the line of the syzygies. 

239. Now any cause which, at the perigee, should have 
the effect to increase the moon's gravitation toward the earth 
beyond its mean, and, at the apogee, to diminish the moon's 
gravitation toward the earth, would augment the difference 



LUNAR IRREGULARITIES. 139 

between the gravitation at the perigee and at the apogee, and 
consequently increase the eccentricity of the orbit. Again, 
any cause which at the perigee should have the effect to lessen 
the moon's gravitation toward the earth, and at the apogee 
to increase it, would lessen the difference between the two, and 
consequently diminish the eccentricity of the orbit, or bring it 
nearer to a circle. Let us see if the disturbing force of the sun 
produces these effects. The sun's disturbing force, as we have 
seen in article 228, admits of two resolutions, one in the direc- 
tion of the radius vector (OM, Fig. 48), the other (OE") in the 
direction of a tangent to the orbit. First, let AB be the line 
of the apsides in syzygy, A being the place of the perigee. The 
sun's disturbing force OM is greatest in the direction of the 
line of the syzygies ; yet depending as it does on the unequal 
action of the sun upon the earth and the moon, and being 
greater as their distance from each other is greater, it is at a 
minimum when acting at the perigee, and at a maximum when 
acting at the apogee. Hence its effect is to draw away the 
moon from the earth less than usual at the perigee, and of 
course to make her gravitation toward the earth greater than 
usual; while at the apogee its effect is to diminish the tendency 
of the moon to the earth more than usual, and thus to increase 
the disproportion between the two distances of the moon from 
the focus at these two points, and of course to increase the 
eccentricity of the orbit. The moon, therefore, if moving 
toward the perigee, is brought to the line of the apsides in a 
point between its mean place and the earth ; or if moving to- 
ward the apogee, she reaches the line of the apsides in a point 
more remote from the earth than its mean place. 

Secondly, let CD be the line of apsides in quadrature. The 
effect of the sun's action is to increase the moon's tendency 
to the earth, when in quadrature. If, however, the moon is 
then at perigee, such increase will be less than usual, because 
the sun's action is less oblique ; and if at apogee, it will be 
more than usual, because more oblique. Hence its effect will 
be to lessen the disproportion between the two distances of the 
moon from the focus at these two points ; and of course to di- 
minish the eccentricity of the orbit, The moon, therefore, if 
moving toward the perigee, meets the line of the apsides in a 
point more remote from the earth than the mean place of the 



140 THE MOON. 

perigee ; and if moving toward the apogee, in a point between 
the earth and the mean place of the apogee.* 

240. A third inequality in the lunar motions, is the Vari- 
ation. By comparing the moon's place as computed from her 
mean motion corrected for the equation of the center and for 
evection, with her place as determined by observation, Tycho 
Brahe discovered that the computed and observed places did 
not always agree. They agreed only in the syzygies and quad- 
ratures, and disagreed most at a point half way between these, 
that is, at the octants. Here, at the maximum, it amounted to 
more than half a degree (35' 41 ".6). It appeared evident from 
examining the daily observations while the moon is perform- 
ing her revolution around the earth, that this inequality is 
connected with the angular distance of the moon from the sun, 
and in every part of the orbit could be correctly expressed by 
multiplying the maximum value, as given above, into the sine 
of twice the angular distance between the sun and the moon. 
It is, therefore, at the conjunctions and quadratures, and 
greatest at the octants. Tycho Brahe knew the fact: Newton 
investigated the cause. 

It appears by article 228, that the sun's disturbing force can 
be resolved into two parts : one in the direction of the radius 
vector, the other at right angles to it in the direction of a tan- 
gent to the moon's orbit. The former, as already explained, 
produces the Evection ; the latter produces the Variation. 
This latter force will accelerate the moon's velocity, in every 
point of the quadrant which the moon describes in moving 
from quadrature to conjunction, or from C to A (Fig. 48), but 
at an unequal rate, the acceleration being greatest at the 
octant, and nothing at the quadrature and the conjunction ; 
and when the moon is past conjunction, the tangential force 
will change its direction and retard the moon's motion. All 
these points will be understood by inspection of figure 48. 

241. A fourth lunar inequality is the Annual Equation. 
This depends on the distance of the earth (and of course the 
moon) from the sun. Since the disturbing influence of the sun 

* Woodhonse's Ast., p. 680. 



LUNAR IRREGULARITIES. 141 

has a greater effect in proportion as the sun is nearer,* conse- 
quently all the inequalities depending on this influence must 
vary at different seasons of the year. Hence, the amount of 
this effect due to any particular time of the year is called the 
Annual Equation. 

242. The foregoing are the largest of the inequalities of the 
moon's motions, and may serve as specimens of the corrections 
that are to be applied to the mean place of the moon in order 
to find her true place. These were first discovered by actual 
observation ; but a far greater number, though most of them 
are exceedingly minute, have been made known by the inves- 
tigations of Physical Astronomy, in following out all the con- 
sequences of universal gravitation. In the best tables, about 
30 equations are applied to the mean motions of the moon. 
That is, we first compute the place of the moon on the supposi- 
tion that she moves uniformly in a circle. This gives us her 
mean place. We then proceed, by the aid of the Lunar Ta- 
bles, to apply the different corrections, such as the equation of 
the center, evection, variation, the annual equation, and so on, 
to the number of 28. Numerous as these corrections appear, 
yet La Place informs us, that the whole number belonging to 
the moon's longitude is no less than 60 ; and that to give the 
tables all the requisite degree of precision, additional investi- 
gations will be necessary, as extensive at least as those already 
made.f The best tables in use in the time of Tycho Brahe, 
gave the moon's place only by a distant approximation. The 
tables in use in the time of Newton (Halley's tables), approxi- 
mated within 7 minutes. Tables at present in use give the 
moon's place to 5 seconds. These additional degrees of accu- 
racy have been attained only by immense labor, and by the 
united efforts of Physical Astronomy and the most refined ob- 
servations. 

243. The inequalities of the moon's motions are divided 
into periodical and secular. Periodical inequalities are those 
which are completed in comparatively short periods, like evec- 



* Varying reciprocally as the cube of the sun's distance from the earth, 
f Syst. du Monde., 1. iv., c. 5. 



142 THE MOON. 

tion and variation ; Secular inequalities are those which are 
completed only in very long periods, such as centuries or ages. 
Hence the corresponding terms periodical equations, and secu- 
lar equations. As an example of a secular inequality, we may 
mention the acceleration of the moon's mean motion. It is dis- 
covered that the moon actually revolves around the earth in 
less time now than she did in ancient times. The difference, 
however, is exceedingly small, being only about 10" in a cen- 
tury, but increases from century to century as the square of 
the number of centuries from a given epoch. This remarkable 
fact was discovered by Dr. Halley.* In a lunar eclipse the 
moon's longitude differs from that of the sun, at the middle of 
the eclipse, by exactly 180° ; and since the sun's longitude at 
any given time of the year is known, if we can learn the day 
and hour when an eclipse occurs, we shall of course know the 
longitude of the sun and moon. Eow in the year 721 before 
the Christian era, on a specified day and hour, Ptolemy records 
a lunar eclipse to have happened, and to have been observed 
by the Chaldeans. The moon's longitude, therefore, for that 
time, is known ; and if we compute the place which the moon 
would now occupy, had she always maintained her present 
period of revolution, we shall find that place to be about a 
degree and a half in advance of her actual place, showing that 
the period we have used is too short. Moreover, the same 
conclusion is derived from a comparison of the Chaldean ob- 
servations with those made by an Arabian astronomer of the 
tenth century. 

This phenomenon at first led astronomers to apprehend that 
the moon encountered a resisting medium, which, by destroy- 
ing at every revolution a small portion of her projectile force, 
would have the effect to bring her nearer and nearer to the 
earth, and thus to augment her velocity. But in 1TS6, La 
Place demonstrated that this acceleration is one of the legiti- 
mate effects of the sun's disturbing force, and is so connected 
with changes in the eccentricity of the earth's orbit, that the 
moon will continue to be accelerated while that eccentricity 
diminishes ; but when the eccentricity has reached its minimum 
(as it will do after many ages), and begins to increase, then 

* Astronomer Royal of Great Britain, and cotemporary with Sir Isaac Newton. 



ECLIPSES. Ii3 

the moon's motion will begin to be retarded, and thus her 
mean motions will oscillate forever about a mean value. 

244. The lunar inequalities which have been considered 
are such only as affect the moon's longitude ; but the sun's 
disturbing force also causes inequalities in the moon's latitude 
and parallax. Those of latitude alone, require no less than 
twelve equations. Since the moon revolves in an orbit in- 
clined to the ecliptic, it is easy to see that the oblique action 
of the sun must admit of a resolution into two forces, one of 
which being perpendicular to the moon's orbit, must effect 
changes in her latitude. Since, also, several of the inequalities 
already noticed, involve changes in the length of the radius 
vector, it is obvious that the moon's parallax must be subject 
to corresponding perturbations. 



CHAPTER VII. 

ECLIPSES. 

245. An eclipse of the moon happens when the moon, in its 
revolution about the earth, falls into the earth's shadow. An 
eclipse of the sun happens when the moon, coming between 
the earth and the sun, covers either a part or the whole of the 
solar disk. An eclipse of the sun can occur only at the time 
of conjunction, or new moon ; and an eclipse of the moon only 
at the time of opposition, or full 'moon. "Were the moon's 
orbit in the same plane with that of the earth, or did it coin- 
cide with the ecliptic, then an eclipse of the sun would take 
place at every conjunction, and an eclipse of the moon at every 
opposition ; for as the sun and earth both lie in the ecliptic, 
the shadow of the earth must also extend in the same piano, 
being, of course, always directly opposite to the sun ; and since, 
as we shall soon see, the length of this shadow is much greater 
than the distance of the moon from the earth, the moon, if it 
revolved in the plane of the ecliptic, must pass through the 



14:4 THE MOON. 

shadow at every full moon. For similar reasons, the moon 
would occasion an eclipse of the sun, partial or total, in some 
portions of the earth at every new moon. But the lunar orbit 
is inclined to the ecliptic about 5°, so that the center of the 
moon, when she is furthest from her node, is 5° from the axis 
of the earth's shadow (which is always in the ecliptic) ; and, as 
we shall show presently, the greatest distance to which the 
shadow extends on each side of the ecliptic, that is, the greatest 
semi-diameter of the shadow, where the moon passes through 
it, is only about f of a degree, while the semi-diameter of the 
moon's disk is only about \ of a degree ; hence the two semi- 
diameters, namely, that of the moon and the earth's shadow, 
cannot overlap one another, unless, at the time of new or full 
moon, the sun is at or very near the moon's node. In the 
course of the sun's apparent revolution around the earth once 
a year, he is successively in every part of the ecliptic ; conse- 
quently, the conjunctions and oppositions of .the sun and moon 
ma}^ occur at any part of the ecliptic, either when the sun is 
at the moon's node (or when he is passing that point of the 
celestial vault on which the moon's node is projected as seen 
from the earth), or they may occur when the sun is 90° from 
the moon's node, where the lunar and solar orbits are at the 
greatest distance from each other ; or, finally, they may occur 
at any intermediate point. Now the sun, in his annual revo- 
lution, passes each of the moon's nodes on opposite sides of the 
ecliptic, and of course at opposite seasons of the year ; so that, 
for any given year, the eclipses commonly happen in two op- 
posite months, as January and July, February and August, 
May and November. These are called Node Months, and 
become earlier each year, because the nodes retrograde. • 

* 
246. If the sun were of the same size with the earth, the 
shadow of the earth would be cylindrical and infinite in length, 
since the tangents drawn from the sun to the earth (which 
form the boundaries of the shadow) would be parallel to each 
other ; but as the sun is a vastly larger body than the earth, 
the tangents converge and meet in a point at some distance 
behind the earth, forming a cone, of which the earth is the 
base, and whose vertex (and of course its axis) lies in the eclip- 
tic. A little reflection will also show us that the form and 



ECLIPSES. 145 

dimensions of the shadow must be affected by several circum- 
stances ; that the shadow must be of the greatest length and 
breadth when the sun is furthest from the earth ; that its 
figure will be slightly modified by the spheroidal figure of the 
earth $ and that the moon, being, at the time of its opposition, 
sometimes nearer to the earth, and sometimes further from it, 
will accordingly traverse it at points where its breadth varies 
more or less, 

247, The semi-angle of the cone of the earth's shadow is 
equal to the sun's apparent semi-diameter, minus his horizontal 
parallax. 

Let AS (Fig. 51) be the semi-diameter of the sun, BE 
that of the earth, and EC the axis of the earth's shadow. 
Then the semi-angle of the cone of the earth's shadow 

Fig. 51. 




ECB = AES — EAB, of which AES is the sun's semi-diame- 
ter, and EAB his horizontal parallax ; and as both these quan- 
tities are known, hence the angle at the vertex of the shadow 
becomes known. Putting 6 for the sun's semi-diameter, and p 
for his horizontal parallax, we have the semi-angle of the earth's 
shadow ECB = 6— p. 

248. At the mean distance of the earth from the sun, the 
length of the earth's shadow is about 860,000 miles, or more 
than three times the distance of the moon from the earth. 

In the right-angled triangle ECB, right angled at B, the 
angle ECB being known, and the side EB, we can find the side 

EC. For sin (3 -p) : EB : : B : EC = ^-=P . This value 

x sm (0 —p) 

will vary with the sun's semi-diameter, being greater as that 

is less. Its mean value being 16' 1".5, and the sun's horizon- 

1 10 



146 THE MOON. 

tal parallax being 8". 6, d — p = 15' 5$,'^ and EB = 3956.2 
Hence, 

Sin 15' 53" : Kad : : 3956.2 : 856,275. 
Since the distance of the moon from the earth is 238,545 
miles, the shadow extends about 3.6 times as far as the moon, 
and consequently the moon passes the shadow toward its broad- 
est part, where its breadth is much more than sufficient to 
cover the moon's disk. 

249. The average breadth of the earth's shadow, where it 
eclipses the moon, is almost three times the moon's diameter. 

Let mm' (Fig. 51) represent a section of the earth's shadow 
where the moon passes through it, M being the center of the 
circular section. Then the angle MEm will be the angular 
breadth of half the shadow. But, 

ME??z = BwE — BCE ; that is, since J$mE is the moon's hori- 
zontal parallax (Art. 82), and BCE equals the sun's semi- 
diameter minus his horizontal parallax {6 — p), therefore, put- 
ting P for the moon's horizontal parallax, we have 

MEm = P - (d -p) = V+p - d ; that is, since P= 57' 1", and 
d-p = 15' 52".9, therefore 57' 1"- 15' 52".9 =41' 8".l, which 
is nearly three times 15' 33", the semi-diameter of the moon. 
Thus it is seen how, by the aid of geometry, we learn to esti- 
mate various particulars respecting the earth's shadow, by 
means of simple data derived from observation. 

250. The distance from the node to the center of the earth's 
shadow, when so situated that the moon would merely touch 
it at opposition, is called the lunar ecliptic limit; and the solar 
ecliptic limit is the distance from the node to the center of the 
sun's disk, when the moon apparently touches it in passing the 
conjunction. All eclipses of the moon and sun occur within 
these limits respectively. 

251. The Lunar Ecliptic Limit is nearly 12 degrees. 

Let CN (Fig. 52) be the sun's path, MN the moon's, and N 
the node. Let Qa be the semi-diameter of the earth's shadow, 
and Ma the semi-diameter of the moon. Since Ca and Ma are 
known quantities, their sum CM is also known. The angle at 
!N" is known, being the inclination of the lunar orbit to the 



ECLIPSES. 147 

ecliptic. Hence, in the spherical triangle MCIST, right angled 
at M,* by Xapier's theorem (Art. 132, Note), 

EadXsin CM=sin CNxsin MNC. 
Fig. 52. 




The greatest apparent semi-diameter of the earth's shadow 
where the moon crosses it, computed by article 249, is 45' 52", 
and the moon's greatest apparent semi-diameter is 16' 45". 5, 
which together give MC equal to 62' 37".5. Taking the incli- 
nation of the moon's orbit, or the angle MNC (what it gener- 
ally is in these circumstances) at 5° 17', and we have Had x sin 

™> r aM , * • ™ T • * Q *», • ™t Radxsin 62' 37".5 

62' 37 / .5=sm CNxsin 5° 17', or sin CN= KQ . - , 

sin 5 17' 

and GN"=11° 25' 40". f This is the greatest distance of the 
moon from her node at which an eclipse of the moon can take 
place. By varying the value of CM, corresponding to varia- 
tions in the distances of the sun and moon from the earth, it is 
found that if NC is less than 9°, there must be an eclipse ; but 
between this and the limit, the case is doubtful. 

When the moon's disk only comes in contact with the earth's 
shadc.w, as in figure 52, the phenomenon is called an appulse / 
when only a part of the disk enters the shadow, the eclipse is 
said to be partial, and total if the whole of the disk enters the 
shadow. The eclipse is called central when the moon's center 
coincides with the axis of the shadow, which happens when the 
moon at the moment of opposition is exactly at her node. 

252. Before the moon enters the earth's shadow, the earth 
begins to intercept from it portions of the sun's light, gradu- 
ally increasing until the moon reaches the shadow. This par- 
tial light is called the moon's Penumbra. Its limits are ascer- 

* The line CM is to be regarded as the projection of the line which connects 
the centers of the moon and section of the earth's shadow, as seen from the 
earth. 

f Woodhouse"s Astronomy, p. 718. 



148 THE MOON. 

tained by drawing the tangents AC'B' and A'C'B. (Fig. 51.) 
Throughout the space included between these tangents, more 
or less of the sun's light is intercepted from the moon by the 
interposition of the earth ; for it is evident, that as the moon 
moves toward the shadow, she would gradually lose the view 
of the sun, until, on entering the shadow, the sun would be 
entirely hidden from her. . 

253. The semi-angle of the Penumbra equals the sun's semi- 
diameicr and horizontal parallax, or d+p. 

The angle ACM (Fig. 51) = AC'S= AES+B'AE. But AES 
is the sun's semi-diameter, and B'AE is the sun's horizontal 
parallax, both of which quantities are known. 

254. The semi-angle of a section of the Penumbra, where 
the moon crosses it, equals the moon's horizontal parallax, plus 
the sun's, plus the sun's semi-diameter. 

The angle AEM (Fig. 51) = EAC'+EC'A. But EAC'=P, 
the moon's horizontal parallax, and EG'h=d-{-p (Art. 253), .\ 
7iEM=P-ri?+#, a ^ which are likewise known quantities. 

Hence, by means of these few elements, which are known 
from observation, we ascend to a complete knowledge of all the 
particulars necessary to be known respecting the moon T s pe- 
numbra. 

255. In the preceding investigations, we have supposed that 
the cone of the earth's shadow is formed by lines drawn from 
the sun, and touching the earth's surface. But the apparent 
diameter of the shadow is found by observation to be somewhat 
greater than would result from this hypothesis. The fact is 
accounted for by supposing that a portion of the solar rays 
which graze the earth's surface are absorbed and extinguished 
by the lower strata of the atmosphere. This amounts to the 
same thing as though the earth were larger than it is, in which 
case the moon's horizontal parallax would be increased ; and 
accordingly, in order that theory and observation may coincide, 
it is found necessary to increase the parallax by -£$. 

256. In a total eclipse of the moon, its disk is still visible, 
shining with a dull red light. This is due to the earth's at- 



ECLIPSES. 



149 



mosphere, which acts as a convex lens, and converges the light 
into the shadow. The lower rays, if they could escape, would 
be bent twice 34' (Art 89), and reach the axis thousands of 
miles this side of the moon. As it is, only a little light emerges, 
which is sufficiently bent to fall on the moon when centrally 
eclipsed. An observer at the moon, in witnessing a solar eclipse, 
would see the sun expanded into a dim narrow ring, having 
nearly four times its usual diameter. 

257. In calculating an eclipse of the moon, we first learn 
from the tables in what month the sun, at the time of full 
moon in that month, is near the moon's node, or within the 
lunar ecliptic limit. This it must evidently be easy to deter- 
mine, since the tables enable us to find both the longitudes of 
the nodes, and the longitudes of the sun and moon, for every 
day of the year. Consequently, we can find when the sun has 
nearly the same longitude as one of the nodes, and also the 
precise moment when the longitude of the moon is 180° from 
that of the sun, for this is the time of opposition, from which 
may be derived the time of the middle of the eclipse. Having 
the time of the middle of the eclipse, and. the breadth of the 
shadow (Art. 249), and knowing, from the tables, the rate at 
which the moon moves per hour faster than the shadow, we 
ean find how long it will take her to traverse half the breadth 
of the shadow ; and this time subtracted from the time of the 
middle of the eclipse, will give the beginning, and added to the 
time of the middle will give the end of the eclipse. Or if in- 
stead of the -breadth of the shadow, we employ the breadth of 
the penumbra {Art. 253), we may find, in the same manner, 
when the moon enters and when she leaves the penumbra. 
We see, therefore, how, by having a few things known by ob- 
servation, such as the sun and moon's semi-diameters, and their 
horizontal parallaxes, we rise, by the aid of trigonometry, to 
the knowledge of various particulars respecting the length and 
breadth of the shadow and of the penumbra. These being 
known, we next have recourse to the tables which contain all 
the necessary particulars respecting the motions of the sun and 
moon, together with equations or corrections, to be applied for 
all their irregularities. Hence it is comparatively an easy task 
to calculate with great accuracy an eclipse of the moon. 



150 THE MOON. 

258. Let us then see how we may find the exact time of the 
beginning, end, duration, and magnitude, of a lunar eclipse. 

Let NG (Fig. 53) be the ecliptic, and ISag the moon's orbit, 
the sun being in A* when the moon is in opposition at a ; let 
N be the ascending node, and Aa the moon's latitude at the 

Fig. 53. 



instant of opposition. An hour afterward the sun will have 
passed to A', and the moon to g, when the difference of longi- 
tude of the two bodies will be GA'. Then gh is the moon's 
hourly motion in latitude, and ah her hourly motion in longi- 
tude. As the character and form of the eclipse will depend 
solely upon the distances between the centers of the sun and 
moon, that is, upon the line gA', instead of considering the two 
bodies as both in motion, we may suppose the sun at rest in A, 
and the moon as advancing with a motion equal to the differ- r 
ence between its rate and that of the sun, a supposition that 
will simplify the calculation. Therefore, draw gd parallel and 
equal to A 7 A, join dA, and this line being equal to gA\ the 
two bodies will be in the same relative situation as if the sun 
were at A' and the moon at g. Join da and produce the line 
da both ways, cutting the ecliptic in F ; then daF will be the 
moon's Relative Orbit. Hence ai — ah— AA'=the difference 
of the hourly motions of the sun and moon, that is, the moon's 
relative motion in longitude, and di=the moon's hourly motion 
in latitude. 

Draw CD (Fig. 54) to represent the ecliptic, and let A be 
the place of the sun. As the tables give the computation of 
the moon's latitude at every instant, consequently, we may 
take from them the latitude corresponding to the instant of op- 
position, and to one hour later ; and we may take also the sun's 
and moon's hourly motions in longitude. Take AD, AB, each 

* It will be remarked that the point A really represents the center of the earth's 
shadow ; but as the real motions of the shadow are the same with the assumed 
motions of the sun, the latter are used in conformity with the language of the 
tables. 



ECLIPSES. 



151 



equal to the relative motion, and Aa—the latitude in opposition, 
Dd=th.e latitude one hour afterward ; join da and produce 
the line da both ways, and it will represent the moon's relative 
orbit. Draw Bb at right angles to CD, and it will be the lati- 

Fig. 54. 




D F 



tude an hour before opposition. At the time of the eclipse, 
the apparent distance of the center of the shadow from the 
moon is very small ; consequently, CD, cd, T>d, &c, may be 
regarded as straight lines. During the short interval between 
the beginning and end of an eclipse, the motion of the sun, and 
consequently that of the center of the shadow, may likewise 
be regarded as uniform. 

259. The various particulars that enter into the calculation 
of an eclipse are called its Elements / and as our object is 
here merely to explain the method of calculating an eclipse of 
the moon (referring to the Supplement for the actual compu- 
tation), we may take the elements at their mean value. Thus, 
we will -consider cd as inclined to CD 5° 9', the moon's hori- 
zontal parallax as 58', its semi-diameter as 16', and that of the 
earth's shadow as 42'. The line Am, perpendicular to cd, gives, 
the point m for the place of the moon at the middle of the 
eclipse, for this line bisects the chord, which represents the 
path of the moon through the shadow ; and mM, perpendicu- 
lar to CD, gives AM for the time of the middle of the eclipse 
before opposition, the number of minutes before opposition 
being the same part of an hour that AM is of AB.* From 
the center A, with a radius equal to that of the earth's shadow 
(42'), describe the semicircle BLF, and it will represent the 



* The situation of the moon when at m is called orbit opposition ; and her situa- 
tion when at a. ecliptic opposition. 



152 THE MOON. 

projection of the shadow traversed by the moon. "With a 
radius equal to the semi-diameter of the shadow and that of 
the moon (=42 '+16' =58'), and with the center A, mark the 
two points o and f on the relative orbit, and they will be the 
places of the center of the moon at the beginning and end of 
the eclipse. The perpendiculars cG, f¥, give the times AC 
and AF of the commencement and the end of the eclipse, and 
CM or MF gives half the duration. From the centers e andy, 
with a radius equal to the semi-diameter of the moon (16'), de- 
scribe circles, and they will each touch the shadow (Euc, 3, 12), 
indicating the position of the moon at the beginning and end 
of the eclipse. If the same circle described from m is wholly 
within the shadow, the eclipse will be total; if it is only partly 
within the shadow, the eclipse will be partial. With the 
center A, and radius equal to the semi-diameter of the shadow 
minus that of the moon (42'— 16' =26'), mark the two points e' y 
f\ which will give the places of the center of the moon, at the 
beginning and end of total darkness, and MC, MF' will give 
the corresponding times before and after the middle of the 
eclipse. Their sum will be the duration of total darkness. 

260. If the foregoing projection be accurately made from 
a scale, the required particulars of the eclipse may be ascer- 
tained by measuring, on the same scale, the lines which re- 
spectively represent them ; and we should thus obtain a near 
approximation to the elements of the eclipse. A more accu- 
rate determination of these elements may', however, be ob- 
tained by actual calculation. The general principles of the 
, calculation will be readily understood. 

First, knowing ai (Fig. 53), the moon's relative longitude, 
and di, her latitude, we find the angle dai, which is the in- 
clination of the moon's relative orbit. But dai=aAm ; and, 
in the triangle «Am, we have the angle at A, and the side A#, 
being the moon's latitude at the time of opposition, which is 
given by the tables. Hence we can find the side Am. In the 
triangle AmM (Fig. 54), having the side Am and the angle 
AmM (=aAm), we can find AM=the arc of relative longitude 
described by the moon from the time of the middle of the 
eclipse to the time of opposition ; and knowing the moon's 
hourly motion in longitude, we can convert AM into time, and 



ECLIPSES. 106 

this subtracted from the time of opposition gives us the time of 
the middle of the eclipse. 

Secondly, joining Ac (Fig. 54), not represented in the fig- 
ure, we calculate mAc, in the triangle Acm ; then mAB (= 
maA)— mAc=cAC. Hence, in the triangle AC<?, we can de- 
termine AC,' and therefore AC — AM=MC, which, changed 
into time as before, gives us, when subtracted from the time 
of the middle of the eclipse, the time of the beginning of the 
eclipse, or, when added to that of the middle, the time of the 
end of the eclipse. The sum of the two equals the whole dura- 
tion. 

Thirdly, by a similar method we calculate the value of MC, 
which converted into time, and subtracted from the time of the 
middle of the eclipse, gives the commencement of total dark- 
ness, or when added gives the end of total darkness. Their 
sum is the duration of total darkness. 

Fourthly, the quantity of the eclipse is determined by sup- 
posing the diameter of the moon divided into twelve equal 
parts called Digits, and finding how many such parts lie 
within the shadow, at the time when the centers of the moon 
and the shadow are nearest to each other. Even when the 
moon lies wholly within the shadow, the quantity of the 
eclipse is still expressed by the number of digits contained in 
that part of the line which joins the center of the shadow and 
the center of the moon, which is intercepted between the edge 
of the shadow and the inner edge of the moon. Thus in figure 

54, the number of digits eclipsed, equals -j — = — r — — = 

Ao—(Am—nm) . ' . . , . 

! — 7 , an expression containing only known quan- 

i^no 

tities. 

261. The foregoing will serve as an explanation of the gen- 
eral principles, on which proceeds the calculation of a lunar 
eclipse. The actual methods practiced employ many expedi- 
ents to facilitate the process, and to insure the greatest possible 
accuracy, the nature of which are explained and exemplified 
in Mason's Supplement to this work. 

262. The leading particulars respecting an Eclipse of the 



154: THE MOON. 

Sun, are ascertained very nearly like those of a lunar eclipse. 
The shadow of the moon travels over a portion of the earth, as 
the shadow of a small cloud, seen from an eminence in a clear 
day, rides along over hills and plains. Let us imagine our- 
selves standing on the moon ; then we shall see the earth par- 
tially eclipsed by the shadow of the moon, in the same manner 
as we now see the moon eclipsed by the earth's shadow ; and 
we might proceed to find the length of the shadow, its breadth 
where it eclipses the earth, the breadth of the penumbra, and 
its duration and quantity, in the same way as we have ascer- 
tained these particulars for an eclipse of the moon. 

But, although the general characters of a solar eclipse might 
be investigated on these principles, so far as respects the earth 
at large, yet as the appearances of the same eclipse of the sun 
are very different at different places on the earth's surface, it 
is necessary to calculate its peculiar aspects for each place sep- 
arately, a circumstance which makes the calculation of a solar 
eclipse much more complicated and tedious than of an eclipse 
of the moon. The moon, when she enters the shadow of the 
earth, is deprived of the light of the part immersed, and that 
part appears black alike to all places where the moon is above 
the horizon. But it is not so with a solar eclipse. We do not 
see this by the shadow cast on the earth, as we should do if 
we stood on the moon, but by the interposition of the moon 
between us and the sun ; and his edge may be hidden from one 
observer while he is in full view of another only a few miles 
distant. In strictness, the phenomenon should be called an 
occultation, not an eclipse, of the sun ; the earth is eclipsed, or 
obscured by a shadow cast upon it, while the sun, to those 
within the shadow, is hidden by an intervening body. 

263. We have compared the motion of the moon's shadow 
over the surface of the earth to that of a cloud ; but its velocity 
is incomparably greater. The mean motion of the moon 
around the earth is about 33' per hour ; but 33' of the moon's 
orbit is 2280 miles, and the shadow moves of course at the 
same rate, or 2280 miles per hour, traversing the entire disk of 
the earth in less than four hours. This is the velocity of the 
shadow when it passes perpendicularly over the earth ; when 
the direction of the axis of the shadow is oblique to the earth's 



ECLIPSES. 155 

surface, the velocity is increased in proportion of radius to the 
sine of obliquity. Thus the shadows of evening have a far 
more rapid motion than those of noonday. 

Let us endeavor to form a just conception of the manner in 
which these three bodies — the sun, the earth, and the moon — are 
situated with respect to each other at the time of a solar 
eclipse. First, suppose the conjunction to take place at the 
node. Then the straight line which connects the centers of 
the sun and the earth, also passes through the center of the 
moon, and coincides with the axis of its shadow ; and, since 
the earth is bisected by the plane of the ecliptic, the shadow 
would traverse the earth in the direction of the terrestrial 
ecliptic, from west to east, passing over the middle regions of 
the earth. Here the diurnal motion of the earth being in the 
same direction with the shadow, but with a less velocity, the 
shadow will appear to move with a speed equal only to the 
difference between the two. Secondly, suppose the moon is 
on the north side of the ecliptic at the time of conjunction, and 
moving toward her descending node, and that the conjunction 
takes place just within the solar ecliptic limit, say 16° from 
the node. The shadow will now not fall in the plane of the 
ecliptic, but a little northward of it, so as just to graze the 
earth near the pole of the ecliptic. The nearer the conjunc- 
tion comes to the node, the further the shadow will fall from 
the pole of the ecliptic toward the equatorial regions. In cer- 
tain cases, the shadow strikes beyond the pole of the earth ; 
and then its easterly motion being opposite to the diurnal mo- 
tion of the places which it traverses, consequently its velocity 
is greatly increased, being equal to the sum of both. 

264. After these general considerations, we will now exam- 
ine more particularly the method of investigating the elements 
of a solar eclipse. 

The length of the moon's shadow is the first object of in- 
quiry. The moon, as well as the earth, is at different distan- 
ces from the sun at different times, and hence the length of 
her shadow varies, being always greatest when she is furthest 
from the sun. Also, since her distance from the earth varies, 
the section of the moon's shadow made by the earth, is greater 
in proportion as the moon is nearer the earth. The greatest 



156 



THE MOON. 



eclipses of the sun, therefore, happen when the sun is in apo- 
gee,* and the moon in perigee. 

265. When the moon is at her mean distance from the 
earth, and from the sun, her shadow nearly reaches the earth's 
surface. 

Let S (Fig. 55) represent the sun, D the moon, and T the 
earth. Then, the semi-angle of the cone of the moon's shadow, 

Fig.55 . 




DKR, will, as in the case of the earth (Art. 247), equal SDR— 
DRK, of which SDR is the sun's apparent semi-diameter, as 
seen from the moon, and DRK, is the sun's horizontal parallax 
at the moon. Since, on account of the great distance of the 
sun compared with that of the moon, the semi-diameter of the 
sun as seen from the moon must evidently be very nearly the 
same as when seen from the earth, and since, on account of the 
minuteness of the moon's semi-diameter when seen from the 
sun, the sun's horizontal parallax at the moon must be very 
small, we might, without much error, take the sun's apparent 
semi-diameter from the earth, as equal to the semi-angle of the 
cone of the moon's shadow ; but, for the sake of greater accu- 
racy, let us estimate the value of the sun's semi-diameter and 
horizontal parallax at the moon. 
Now, SDR: STR:: ST : SDf 



|^ STR=1.0025 



: 400 : 399 ; hence SDR= 
STR; and the sun's mean semi-diameter 

16' 



STR being 16.025, hence SDR=1.0025xl6.025 = l6.065 
3".9. 

Again, since parallax is inversely as the distance, the sun's 



« The sun is said to be in apogee, when the earth is in aphelion. 
| The apparent magnitude of an object being reciprocally as its distance from 
the eye. See Note, p. 87. 



ECLIPSES. 157 

horizontal parallax at the moon, is, since she is nearer to the 
snn, about -^ greater than at the earth ; but on account of 
her inferior size it is ^f-£§ less than at the earth. Hence, in- 
creasing the sun's horizontal parallax at the earth by the for- 
mer fraction, and diminishing it by the latter, we have -^-— x 

Out/ 

— — — x9 // =2".5=the sun's horizontal parallax at the moon. 

Therefore, the semi-angle of the cone of the moon's shadow, 
which, as appears above, equals SDR— DEK, equals 16' 3".9 
— 2".5 = 16' 1".4, which so nearly equals the sun's apparent 
semi-diameter, as seen from the earth, that we may adopt the 
latter as the value of the semi-angle of the shadow. Hence, 
sin 16' 1".5 : 1080 (BD) : : Ead : DK=231690. But the 
mean distance of the moon from the surface of the earth is 
238545-3956=234589, which exceeds a little the mean length 
of the shadow as above. 

But when the moon is nearest the earth, her distance from 
the center of the earth is only 221,148 miles ; and when the 
earth is furthest from the sun, the sun's apparent semi-diame- 
ter is only 15' 45". 5. By employing this number in the fore- 
going estimate, we shall find the length of the shadow 235,630 
miles; and 235630-221148=14482, the distance which the 
moon's shadow may reach beyond the center of the earth. 

266. The diameter of the rnoorCs shadow where it traverses 
the earth, i's y at its maximum, about 170 miles* 

In the triangle *TK, the angle at K=15' 45".5 (Art. 265), 
the side Te=3956, and TK= 14482. 

Or, 3956 : 14482 : : sin 15' 45".5 : sin 57' 41".5. 

And 57' 41".5+ 15' 45".5=1° 13' 27"-<ST«, or the arc de. 

And 2de=2° 26' 54"=m. 

Hence 360 : 2.45 (=2° 26' 54") : : 24899f : 170 (nearly). 

26 7. The greatest portion of the earth's surface ever covered 
by the moon 's penumbra, is about 4393 miles. 

c " This supposes the conjunction to take place at the node, and the shadow to 
strike the earth perpendicularly to its surface ; where it strikes obliquely, the 
section may be greater than this. 

f The equatorial circumference. 



158 



THE MOON. 



The semi-angle of the penumbra BID=BSD-f-SBB, of 
which BSD the sun's horizontal parallax at the moon =2". 5, 
and SBR the sun's apparent semi-diameter=16' 3". 9, and 
hence BID is known. The moon's apparent semi-diameter 
BGD=16' 45".5. Therefore GDT is known, as likewise DT 
and TG-. Hence the angle GTd may be found, and the arc dG 
and its double GH, which equals the angular breadth of the 
penumbra. It may be converted into miles by stating a pro- 
portion as in article 266. On making the calculation it will 
be found to be 4393 miles. 

268. The apparent diameter of the moon is sometimes 
larger than that of the sun, sometimes smaller, and sometimes 
exactly equal to it. Suppose an observer placed on the right 
line which joins the centers of the sun and moon ; if the ap- 
parent diameter of the moon is greater than that of the sun, 
the eclipse will be total. If the two diameters are equal, the 
moon's shadow just reaches the earth, and the sun is hidden 
but for a moment from the view of spectators situated in the 

Fig. 55'. 




line which the vertex of the shadow describes on the surface of 
the earth. But if, as happens when the moon comes to her 
conjunction in that part of her orbit which is toward her apo- 
gee, the moon's diameter is less than the sun's, then the ob- 



ECLIPSES, 159 

server will see a ring of the sun encircle the moon, constitu- 
ting an annular eclipse. (Fig. 55'.) 

269. Eclipses of the sun are modified by the elevation of 
the moon above the horizon, since its apparent diameter is 
augmented as its altitude is increased (Art. 217). This effect, 
combined with that of parallax, may so increase or diminish 
the apparent distance between the centers of the sun and 
moon, that from this cause alone, of two observers at a dis- 
tance from each other, one might see an eclipse which was not 
visible to the other. If the horizontal diameter of the moon 
differs but little from the apparent diameter of the sun, the 
case might occur where the eclipse would be annular over the 
places where it was observed morning and evening, but total 
where it was observed at mid-day. 

The earth in its diurnal revolution and the moon's shadow 
both move from west to east, but the shadow moves faster 
than the earth ; hence the moon overtakes the sun on its 
western limb and crosses it from west to east. The excess 
of the apparent diameter of the moon above that of the 
sun in a total eclipse is so small, that total darkness seldom 
continues longer than four minutes, and can never continue 
so long as eight minutes. An annular eclipse may last 
12m. 24s. 

Since the sun's ecliptic limits are more than 17° and the 
moon's less than 12°, eclipses of the sun are more frequent than 
those of the moon. Yet lunar eclipses being visible to every 
part of the terrestrial hemisphere opposite to the sun, while 
those of the sun are visible only to the small portion of the 
hemisphere on which the moon's shadow falls, it happens that 
for any particular place on the earth, lunar eclipses are more 
frequently visible than solar. In any year, the number of 
eclipses of both luminaries can not be less than two nor more 
than seven : the most usual number is four, and it is very rare 
to have more than six. The sun does not remain long enough 
near a node for the moon to be in syzygy, within ecliptic 
limits, more than three times. Hence, only three eclipses can 
occur successively while the sun is near a node. As he passes 
both nodes in the same year, there may therefore be six eclipses. 
But a seventh may possibly come just within 12 months, 



160 THE MOON. 

reckoned from the first, in consequence of the backward mo- 
tion of the nodes. 

270. It has been observed already, that were the spectator 
on the moon instead of on the earth, he would see the earth 
eclipsed by the moon, and the calculation of the eclipse would 
be very similar to that of a lunar eclipse; but to an observer 
on the earth the eclipse does not of course begin when the 
earth first enters the moon's shadow, and it is necessary to de- 
termine not only what portion of the earth's surface will be 
covered by the moon's shadow, but likewise the path described 
by its center relative to various places on the surface of the 
earth. This is known when the latitude and longitude of the 
center of the shadow on the earth is determined for each 
instant, The latitude and longitude of the moon are found on 
the supposition that the spectator views it from the center of 
the earth, whereas his position on the surface changes, in con* 
sequence of parallax, both the latitude and longitude, and the 
amount of these changes must be accurately estimated, before 
the appearance of the eclipse at any particular place can be 
fully determined. 

The details of the method of calculating a solar eclipse can 
not be understood in any way so well as by actually perform- 
ing the process according to a given example. For such 
details, therefore, the reader is referred to the Supplement. 

271. In total eclipses there has sometimes been seen a 
remarkable radiation of light around the moon, while the sun 
is behind it. This is called a corona, and appears to be con- 
centric with the sun's disk, rather than with the moon's. It is 
by some considered to indicate the existence of an extensive 
solar atmosphere. 

Another interesting phenomenon often attends the moment of 
concealment and reappearance of the sun's edge at the begin- 
ning and end of total darkness, as also the formation and rup- 
ture of the ring in annular eclipses. It is the dividing up of 
the fine thread of the sun's edge into a series of bright beads. 
Being first noticed by Mr. Francis Baily, they are known by 
the name of Baily* 8 Beads. The appearance is by some at- 
tributed to the light of the sun's edge coming through between 



LONGITUDE. 161 

the mountain summits of the rough outline of the moon's disk. 
That they are not always seen may arise from the fact that the 
edge may in some cases be much less serrated by mountains 
than in others.* 

There is another peculiarity for which no satisfactory ex- 
planation is yet offered. The moment succeeding the total 
occupation of the sun, irregular projections start out from the 
edge here and there, entirely detached from each other, either 
of a flame or a rose color, sometimes short and wide, at others 
narrow, 2' or 3' long, and often bent at a considerable angle. 
During the continuance of the eclipse, these change their forms, 
or disappear, and new ones start up elsewhere. The instant 
that the thread-like edge of the sun appears, the white corona 
and the flame-colored protuberances vanish. 

A total eclipse of the sun. is one of the most sublime and 
impressive phenomena of nature. The darkness is such that 
the larger planets and stars appear, and a chill is felt like that 
of night. Flowers shut up, and animals retire to rest. It is 
not strange that people of barbarous countries are filled with 
consternation and fear by the occurrence of a total eclipse. 



CHAPTER YIIL 



LONGITUDE TIDES. 



27 2. As eclipses of the sun afford one of the most approved 
methods of finding the longitudes of places, our attention is 
naturally turned next toward that subject. 

The ancients studied astronomy in order that they might 
read their destinies in the stars; the moderns, that they may 
securely navigate the ocean. A large portion of the refined 
labors of modern astronomy has been directed toward perfect- 
ing the astronomical tables, with the view of finding the longi- 
tude at sea — an object manifestly worthy of the highest efforts 



Lardner. 
11 



162 THE MOON. 

of science, considering the vast amount of property and of 
human life involved in navigation. 

273. The difference of longitude between two places, may be 
found by any method by which we can ascertain the difference 
of their local times at the same instant of absolute time. 

As the earth turns on its axis from west to east, any place 
that lies eastward of another will come sooner under the sun, 
or will have the sun earlier on the meridian, and consequently, 
in respect to the hour of the day, will be in advance of the 
other at the rate of one hour for every 15°, or four minutes of 
time for each degree. Thus, to a place 15° east of Greenwich, 
it is 1 o'clock, p. m., when it is noon at Greenwich ; and to a 
place 15° west of that meridian, it is 11 o'clock, a. m. at the 
same instant. Hence, the difference of time at any two places 
indicates their difference of longitude. 

27 4. The easiest method of finding the longitude is by 
means of an accurate timepiece, or chronometer. Let us set 
out from London, with a chronometer accurately adjusted to 
Greenwich time, and travel eastward to a certain place, where 
the time is accurately kept, or may be ascertained by observa- 
tion. We find, for example, that it is 1 o'clock by our chro- 
nometer, when it is 2 o'clock and 30 minutes at the place of 
observation. Hence, the longitude is 15 x 1.5 = 22J E. Had 
we traveled westward until our chronometer was an hour and 
a half in advance of the time at the place of observation (that 
is, so much later in the day), our longitude would have been 
22-^-° W. But it would not be necessary to repair to London in 
order to set our chronometer to Greenwich time. This might 
be done at any observatory, or any place whose longitude had 
been accurately determined. For example, the time at New 
York is 4-h. 56m. 4 s . 5 behind that of Greenwich. If, therefore, 
we set our chronometer so much before the true time at New 
York, it will indicate the time at Greenwich. Moreover, on 
arriving at different places, anywhere on the earth, whose 
longitude is accurately known, we may learn whether our 
chronometer keeps accurate time or not ; and if not, the 
amount of its error. Chronometers have been constructed of 
such an astonishing degree of accuracy, as to deviate but a few 



LONGITUDE. 163 

seconds in a- voyage from London to Baffin's Bay and back, 
during an absence of several years. But chronometers which 
are sufficiently accurate to be depended on for long vo} T ages, 
are too expensive for general use, and the means of verifying 
their accuracy are not sufficiently easy.' Moreover, chronom- 
eters, by being transported from one place to another, change 
their daily rate, or depart from that mean rate which they 
preserve at a fixed, station. A chronometer, therefore, can 
not be relied on for determining the longitudes of places 
where the greatest degree of accuracy is required, especially 
where the instrument is conveyed over land, although the 
uncertainty attendant on one instrument may be nearly ob- 
viated by employing several, and taking their mean results.* 

275. Eclipses of the sun and moon are sometimes used for 
determining the longitude. The exact instant of immersion or 
of emersion, or any other definite moment of the eclipse which 
presents itself to two distant observers, affords the means of 
comparing their difference of time, and hence of determining 
their difference of longitude. Since the entrance of the moon 
into the earth's shadow, in a lunar eclipse, is seen at the same 
instant of absolute time at all places where the eclipse is visible 
(Art. 262), this observation would be a very suitable one for 
finding the longitude, were it not that, on account of the in- 
creasing darkness of the penumbra near the boundaries of the 
shadow, it is difficult to determine the precise instant when the 
moon enters the shadow. By taking observations on the im- 
mersions of known spots on the lunar disk, a mean result may 
be obtained which will give the longitude with tolerable accu- 
racy. In an eclipse of the sun, the instants of immersion and 
emersion may be observed with greater accuracy, although, 
since these do not take place at the same instant of absolute 
time, the calculation of the longitude from observations on a 
solar eclipse is complicated and laborious. 

A method very similar to the foregoing, by observations on 
eclipses of Jupiter's satellites, and on occupations of stars, will 
be mentioned hereafter. 



* Woodhouse, p. 838. 



164 THE MOON. 

276. The Lunar method of finding the longitude, at sea, is 
in man j respects preferable to every other. It consists in 
measuring (with a sextant) the angular distance between the 
moon and the sun, or between the moon and a star, and then 
turning to the Nautical Almanac,* and finding what time it 
was at Greenwich when that distance was the same. The 
moon moves so rapidly, that this distance will not be the same 
except at very nearly the same instant of absolute time. For 
example, at 9 o'clock, a. m., at a certain place, we find the 
angular distance of the moon and the sun to be 72° ; and on 
looking into the Nautical Almanac, we find that the time 
when this distance was the same for the meridian of Greenwich 
was 1 o'clock, p. m. ; hence we infer that the longitude of the 
place is four hours, or 60° west. 

The Nautical Almanac contains the true angular distance of 
the moon from the sun, from the four large planets (Yenus, 
Mars, Jupiter, and Saturn), and from nine bright fixed stars, 
for the beginning of every third hour of mean time for the 
meridian of Greenwich ; and the time corresponding to any 
intermediate distance, may be found by proportional parts.f 

277. It would be a very simple operation to determine the 
longitude by Lunar Distances, if the process, as described in 
the preceding article, were all that is necessary ; but the vari- 
ous circumstances of parallax, refraction, and dip of the hori- 
zon, would differ more or less at the two places, even were the 
bodies (whose distances were taken) in view from both, which is 
not necessarily the case. The observations, therefore, require 
to be reduced to the center of the earth, being cleared of the 
effects of parallax and refraction. Hence, three observers are 
necessary in order to take a lunar distance in the most exact 
manner, viz., two to measure the altitudes of the two bodies 
respectively, at the same time that the third takes the angular 
distance between them. The altitudes of the two luminaries 



* The Nautical Almanac is published annually three or four years in advance, 
containing all necessary tables and information, for the use of navigators. The 
English Board of Longitude have for a long series of years issued such a work. 
The American Ephemeris and Nautical Almanac, which possesses the same general 
character, was commenced but a few years since. 

f See Bowditch's Navigator, tenth ed., p. 226. 



tid.es. 165 

at the time of observation must be known, in order to estimate 
the effects of parallax and refraction. 

278. Since the invention of the magnetic telegraph, it has 
been employed to determine the differences of longitude be- 
tween fixed stations on land with a precision which was before 
altogether unattainable. Suppose two stations to be connected 
by the telegraphic line, and that there is at each a clock keeping 
the local time. The observers agree beforehand at what time 
by his own clock the one at the most easterly station shall com- 
mence giving signals ; and also at what time the other shall 
commence giving another series according to his clock. The 
interval between successive signals is also previously deter- 
mined. When the moment arrives, the first observer strikes 
the telegraphic key at the exact beat of the clock, and the 
seeond observer records the time of the signal as shown by his 
own clock, and thus they continue to do till the full series is 
recorded. The second observer then commences sending sig- 
nals, which are in like manner recorded by the first. The ve- 
locity of the electric current is so great that the absolute time 
of making a signal at one station, and of perceiving it at the 
other, may be considered identical, so that the difference which 
is indicated by the two clocks in each case is wholly due to dif- 
ference of longitude. Still greater precision is attained by 
causing the signal key at each station to record its own move- 
ment on the line of second-marks made by the clock at the- 
other station (Art. 125). 

TIDES. 

279. The tides are an alternate rising and falling of the 
waters of the ocean at regular intervals. They have a maxi- 
mum and minimum twice a day ; and the daily maximum and 
minimum reach their highest and lowef : values twice in a sy- 
nod ical revolution of the moon, or 29^ days. The maximum 
of the daily tide is called high tide, and the minimum low tide. 
The maximum tide during a lunation is called the spring tid*', 
the minimum, neap tide. The rising of the tide is called flood^ 
and the falling, ebb. 

Similar tides, whether high or low, occur on opposite sides 



1G6 THE MOON. 

of the earth at once. Thus at the same time it is high tide at 
any given place, it is also high tide on the inferior meridian ? 
and the same is true of the low tides. 

The interval between two successive high tides is 12h. 25m. ; 
or, if the same tide be considered as returning to the meridian, 
after having gone around the globe, its return is about 50 min- 
utes later than it occurred on the preceding day. In this re- 
spect, as well as in various others, it corresponds very nearly 
to the motions of the moon. 

The average height for the whole globe is about 2^ feet ; or, 
if the earth were covered uniformly with a stratum of water, 
the difference between the two diameters of the oval would be 
5 feet, or more exactly 5 feet and 8 inches ; but its natural 
height at various places is very different, sometimes rising to 60 
or TO feet, and sometimes being scarcely perceptible. At the 
same place, also, the phenomena of the tides are very different 
at different times. 

Inland lakes and seas, even those of the largest class, as Lake 
Superior, or the Caspian, have no perceptible tide. 

280. Tides are caused by the unequal attraction of the sun 
and moon upon different parts of the earth. 

Suppose the projectile force by which the earth is carried 
forward in her orbit to be suspended, and the earth to fall 
toward one of these bodies, the moon for example, in conse- 
quence of their mutual attraction. Then, if all parts of the 
earth fell equally toward the moon, no derangement of its dif- 
ferent parts would result, any more than of the particles of a 
drop of water in its descent to the ground. But if one part 
fell faster than another, the different portions would evidently 
be separated from each other. Now this is precisely what takes 
place with respect to the earth in its fall toward the moon. 
The portions of the earth in the hemisphere next to the moon, 
on account of being n< arer to the center of attraction, fall faster 
than those in the opp( site hemisphere, and consequently leave 
them behind. The solid earth, on account of its cohesion, can 
not obey this impulse, since all its different portions constitute 
one mass, which is acted on in the same manner as though it 
were all collected in the center ; but the waters on the surface, 
moving freely under this impulse, endeavor to desert the solid 



TIDES* 



167 



mass arid fall toward the moon. For a similar reason the wa- 
ters in the opposite hemisphere falling less toward the moon 
than the solid earth, are left behind, or appear to rise from the 
center of the earth. 




281. Let DEFG (Fig. 56) represent the globe; and, for the 
sake of illustrating the principle, we will suppose the waters 
entirely to cover the globe at a uniform depth. Let defg repre- 
sent the solid globe, and the circular 
ring exterior to it, the covering of wa- 
ters. Let C be the center of gravity 
of the solid mass, A that of the hemi- 
sphere next to the moon, and B that 
of the remoter hemisphere. Now the 
force of attraction exerted by the 
moon acts in the same manner as 
though the solid mass were all concen- 
trated in C, and the waters of each 
hemisphere at A and B respectively ; 
and (the moon being supposed above E) it is evident that A 
will tend to leave C, and C to leave B behind. The same must 
evidently be true of the respective portions of matter, of which 
these points are the centers of gravity. The waters of the globe 
will thus be reduced to an oval shape, being elongated in the 
direction of that meridian which is under the moon, and flat- 
tened in the intermediate parts, and most of all at points ninety 
degrees* distant from that meridian. 

Were it not, therefore, for impediments which prevent the 
force from producing its full effects, we might expect to see 
the great tide-wave, as the elevated crest is called, always 
directly beneath the moon, attending it regularly around the 
globe. But the inertia of the waters prevents their instantly 
obeying the moon's attraction, and the friction of the waters on 
the bottom of the ocean, still further retards its progress. It is 
not therefore until several hours (differing at different places) 
after the moon has passed the meridian of a place, that it is 
high tide at that place. 

282. The sun has a similar action to the moon, but only 
one-third as great. On account of the great mass of the sun 



168 THE MOON. 

compared with that of the moon, we might suppose that his 
action in raising the tides would be greater than the moon's ; 
but the nearness of the moon to the earth more than compen- 
sates for the sun's greater quantity of matter. Let us, however, 
form a just conception of the advantage which the moon de- 
rives from her proximity. It is not that her actual amount of 
attraction is thus rendered greater than that of the sun ; but it 
is that her attraction for the different parts of the earth is very 
unequal, while that of the sun is nearly uniform. It is the in- 
equality of this action, and not the absolute force, that pro- 
duces the tides. The diameter of the earth is ^ of the distance 
of the moon, while it is less than T o^o o °f tne distance of the 
sun. 

283. Having now learned the general cause of the tides, 
we will next attend to the explanation of particular phenomena. 

The Spring tides, or those which rise to an unusual height 
twice a month, are produced by the sun and moon's acting in 
a line ; and the Neap tides, or those which are unusually low 
twice a month, are produced by the sun and moon's acting 90 
degrees from each other. The Spring tides occur at the syzygies ; 
the J^eap tides at the quadratures. At the time of new moon, 
the sun and moon both being on the same side of the earth, 
and acting upon it in the same line, their actions conspire, and 
the sun may be considered as adding so much to the force of 
the moon. We have already explained how the moon con- 
tributes to raise a tide on the opposite side of the earth. But 
the sun as well as the moon raises its own tide-wave, which, at 
new moon, coincides with the lunar tide- wave. At full moon 
also, the two luminaries conspire in the same way to raise the 
tide ; for we must recollect that each body contributes to raise 
the tide on the opposite side of the earth as well as on the side 
nearest to it. At both the conjunctions and oppositions, there- 
fore, that is, at thq syzygies, we have unusually high tides. 
But here also the maximum effect is not at the moment of the 
syzygies, but 36 hours afterward. 

At the quadratures, the solar wave is lowest where the lunar 
wave is highest ; hence the low tide produced by the sun is 
subtracted from high water and produces the ]STeap tides. 
Moreover, at the quadratures the solar wave is highest where 



TIDES. 



169 



the lunar wave is lowest, and hence is to be added to the 
height of low water at the time of Neap tides. Hence the 
difference between high and low water is only about half as 
great at Neap tide as at Spring tide. 

284. The power of the moon or of the sun to raise the tide 
is found by the doctrine of universal gravitation to be inversely 
as the cube of the distance* The variations of distance in the 
sun are not great enough to influence the tides very materially, 
but the variations in the moon's distances have a striking 
effect. The tides which happen when the moon is in perigee, 
are considerably greater than when she is in apogee ; and if 
she happens to be in perigee at the time of the syzygies, the 
Spring tide is unusually high. 

285. The declinations of the sun and moon cause the im 
mediate tide, at a given latitude, to be greater than the opposite 
one, or the reverse, according as the declination and latitude 
are alike or unlike. When the declination is nothing, both 
tides are alike at every place. For if the moon is in the plane 





| *© 




of the equator (Fig. 57),f then the highest points of both tide- 
waves are also on the equator; and, at a given latitude, a place 
by the earth's rotation is carried round through T2, T3, at 



° La Place. Syst. du Monde, 1. iv., c. x. 

f Diagrams like these are apt to mislead the learner, by exhibiting the pro- 
tuberance occasioned by the tides as much greater than the reality. We must 
recollect that it amounts, at the highest, to only a very few. feet in eight thou- 
sand miles. Were the diagram, therefore, drawn in just proportions, the altt ra- 
tions of figure produced by the tides would be wholly insensible. 



170 THE MOON. 

equal distances from the highest points, and therefore the im- 
mediate tide T3, and the opposite one T2, are equal ; both are 
less, the higher the latitude, and at the poles there is no tide. 
When the moon is on the north side of the equator, as in fig- 
ure 58, at her greatest northern declination, a place describing 
the parallel TT' will have T'3 for the height of the tide when 
the moon is on the superior meridian, and T2 for the height 
when the moon is on the inferior meridian. Therefore, all 
places north of the equator will have the highest tide when the 
moon is above the horizon, and the lowest when she is below 
it ; the difference of the tides diminishing toward the equator, 
where thej are equal. At the same time, places south of the 
equator have the highest tides w r hen the moon is below the 
horizon, and the lowest when she is above it. "When the moon 
is at her greatest declination, the highest tides will take place 
toward the tropics. The circumstances are all reversed when 
the moon is south of the equator.* 

286. The motion of the tide-wave, it should be remarked, 
is not a progressive motion, but a mere undulation, and is to 
be carefully distinguished from the currents to which it gives 
rise. If the ocean enveloped the earth, and the sun and moon 
were at rest in the equator, the tide-wave would travel at the 
same rate as the earth on its axis. Indeed, the correct way of 
conceiving of the tides, is first to regard the moon as at rest, 
and the earth as revolving and bringing successive parts under 
it, which parts are thus elevated in succession ; and then, to 
consider the moon and tides as moving east 13° per day; thus 
making the time of the relative revolution of the tides west- 
ward, near 25 instead of 24 hours. 

287. The tides of rivers, narrow bays, and shores far from 
the main body of the ocean, are not produced in those places 
by the direct action of the sun and moon, but are subordinate 
waves propagated from the great tide- wave. 

Lines drawn through all the adjacent parts of any tract of 
water, which have high water at the same time, are called co- 
tidal lines.f "We may, for instance, draw a line through all 

* Edin Encyc , Art. Astronomy, p. 623. 

f Whewell, Phil. Transactions for 1833. p 148. 



TIDES. 



171 



places in the Atlantic Ocean which have high tide on a given 
day at 1 o'clock, and another through all places which have 
high tide at 2 o'clock. The cotidal line for any hour may be 
considered as representing the summit or ridge of the tide- 
wave at that time ; and could the spectator, detached from the 
earth, perceive the summit of the wave, he would see it trav- 
eling round the earth in the open ocean once in twenty-five 
hours, followed by another in twelve and a half hours, both 
sending branches into rivers, bays, and other openings into the 
main land. These latter are called Derivative tides, while 
those raised directly by the action of the sun and moon, are 
called Primitive tides. 

288. The velocity with which the wave moves will depend 
on various circumstances, but principally on the depth, and 
probably on the regularity of the channel. If the depth be 
nearly uniform, the cotidal lines will be nearly straight and 
parallel. But if some parts of the channel are deep, while 
others are shallow, the tide will be detained by the greater 
friction of the shallow places, 
and the cotidal lines will be 
irregular. The direction, also, 
of the derivative tide may be 
totally different from that of 
the primitive. Thus (Fig. 59), 
if the great tide- wave, moving 
from east* to west, be repre- 
sented by the lines 1, 2, 3, 4 ; 
the derivative tide, which is 
propagated up a river or bay, 
will be represented by the co- 
tidal lines 3, 4, 5, 6, 7. Ad- f # W 
vancing faster in the channel f * * 
than next the banks, the tides will las: behind toward the 
shores, and the cotidal lines w T ill take the form of curves, as 
represented in the diagram. 



Fig. 59. 




St 3r 



-289. On account of the retarding influence of shoals, and 
an uneven, indented coast, the tide-wave travels more slowly 
along the shores of an island than in the neighboring sea, 



172 THE MOON. 

assuming convex figures at a little distance from the island 
and on opposite sides of it. These convex lines sometimes 
meet and become blended in such a manner as to create singu- 
lar anomalies in a sea much broken by islands, as well as on 
coasts indented with numerous bays and rivers.* Peculiar 
phenomena are also produced, when the tide flows in at oppo- 
site extremities of a reef or island, as into the two opposite 
ends of Long Island Sound. In certain cases a tide-wave is 
forced into a narrow arm of the sea, and produces very re- 
markable tides. The tides of the Bay of Fundy (the highest 
in the world j sometimes rise to the height of 60 or 70 feet ; 
and the tides of the river Severn, near Bristol, in England, 
rise to the height of 40 feet. 

290. The Unit of Altitude of any place is the height of the 
maximum tide after the syzygies (Art. 283), being usually 
about 36 hours after the new or full moon. But as the 
amount of this tide would be affected by the distance of the 
sun and moon from the earth (Art. 284), and by their declina- 
tions (Art. 285), these distances are taken at their mean value, 
and the luminaries are supposed to be in the equator ; the ob- 
servations being so reduced as to conform to these circum- 
stances. The unit of altitude can be ascertained by observa- 
tion only. The actual rise of the tide depends much on the 
strength and direction of the wind. When high winds con- 
spire with a high flood tide, as is frequently the case near the 
equinoxes, the tide rises to a very unusual height. We sub- 
join, from the American Almanac, a few examples of the unit 
of altitude for different places. 

Feet. 

Cumberland, head of the Bay of Fundy, . 71 

Boston, 11| 

New Haven, 8 

New York, .5 

Charleston, S. C, 6 

291. The Establishment of any port is the mean interval 
between noon and the time of high water, on the day of new 

* See an excellent representation and description of these different phenomena 
by Professor Whe well, Phil. Trans., 1833, p. 153. 



TIDES. 173 

or full moon. As the interval for any given place is always 
nearly the same, it becomes a criterion of the retardation of 
the tides at that place. On account of the importance to 
navigation of a correct knowledge of the tides, the British 
Board of Admiralty, at the suggestion of the Eoyal Society, 
recently issued orders to their agents in various important 
naval stations, to have accurate observations made on the 
tides, with the view of ascertaining the establishment and 
various other particulars respecting each station ;* and the 
government of the United States is prosecuting similar investi- 
gations respecting our own ports. 

292. According to Professor Tvnewell,t the tides on the 
coast of North America are derived from the great tide-wave 
of the South Atlantic, which runs steadily northward along 
the coast to the mouth of the Bay of Fundy, where it meets 
the northern tide-wave flowing in the opposite direction. 
Hence he accounts for the high tides of the Bay of Fundy. 

293. The largest lakes and inland seas have no perceptible 
tides. This is asserted by all writers respecting the Caspian 
and Euxine, and the same is found to be true of the largest of 
the North American -Lakes, Lake Superior.^ 

Although these several tracts of water appear large, when 
taken by themselves, yet they occupy but small portions of 
the surface of the globe, as will appear evident from the delin- 
eation of them on an artificial globe. Now we must recollect 
that the primitive tides are produced by the unequal action of 
the sun and moon upon the different parts of the earth ; and 
that it is only at points whose distance from each other bears a 
considerable ratio to the whole distance of the sun or the moon, 
that the inequality of action becomes manifest. The space re- 
quired is larger than either of these tracts of water. It is 
obvious, also, that they have no opportunity to be subject to a 
derivative tide. 

294. To apply the theory of universal gravitation to all 



* Lubbock, Report on the Tides, 1833. f Phil. Trans., 1833, p. 172. 

% See Experiments of Gov. Cass, Am. Jour. Science. 



174: THE PLANETS. 

the varying circumstances that influence the tides, becomes a 
matter of such intricacy, that La Place pronounces " the 
problem of the tides" the most difficult problem of celestial 
mechanics. 

295. The Atmosphere that envelops the earth must evi- 
dently be subject to the action of the same forces as the cov- 
ering of waters, and hence we might expect a rise and fall of 
the barometer, indicating an atmospheric tide corresponding 
to the tide of the ocean. La Place has calculated the amount 
of this aerial tide. It is too inconsiderable to be detected by 
changes in the barometer, unless by the most refined observa- 
tions. Hence it is concluded that the fluctuations produced by 
this cause are too slight to affect meteorological phenomena in 
any appreciable degree.* 



CHAPTEE IX. 

OF THE PLANETS— INFERIOR PLANETS, MERCURY AND VENUS. 

296. The name planet signifies a wanderer^ and is ap- 
plied to this class of bodies because they shift their positions 
in the heavens, whereas the fixed stars apparently always 
maintain the same places with respect to each other. The 
planets known from a high antiquity, are Mercury, Venus, 
Earth, Mars, Jupiter, and Saturn. To these, in 1781, was added 
Uranus X (or H^rschel, as it was formerly called, from the name 
of its discoverer), and, as late as 18-16, another large planet, 
Neptune, was added to the list, making eight in all of those 
bodies usually called planets. Besides these, there is, between 
Mars and Jupiter, a remarkable group of small planets, called 
Asteroids, or more properly, Planetoids. Four of them were 
discovered near the beginning of the present century. From 



* Bowditch's La Place, ii., p. 797. f From the Greek. rAn^r^. 

% From Ovpavot. 



PRIMARY AND SECONDARY PLANETS. 175 

that time till 1845, none were added ; but since 1845, scarce a 
year has passed in which one or more have not been discov- 
ered ; and sometimes five or six have been found in a single 
year. Though names have generally been given them, they 
are more commonly designated by a number, in the order of 
their discovery, the number being inclosed in a circle, which 
is intended to represent the planetary disk. 

The foregoing are called primary planets. Several of these 
have one or more attendants, or satellites, which revolve around 
them as they revolve around the sun. The Earth has one sat- 
ellite, namely, the moon ; Jupiter has four ; Saturn, eight ; 
Uranus, six ;* and Neptune, one. These bodies, also, are 
planets, but in distinction from the others, they are called 
secondary planets. Grouping the planetoids together, and 
giving them the rank of one primary planet (though all united 
would make but a small one), there are nine primaries and 
twenty secondaries. 

297. The primary planets all (with the exception of the 
asteroids) have their orbits nearly in the same plane, and are 
never seen far from the ecliptic. Mercury, whose orbit is most 
inclined of all, never departs further from the ecliptic than 
about 7°, while most of the other planets pursue very nearly 
the same path with the earth, in their annual revolution 
around the sun. The asteroids, however, make wider excur- 
sions from the plane of the ecliptic, amounting, in the case of 
Pallas, to 34^°. 

298. Mercury and Yenus are called inferior planets, be- 
cause their orbits are nearer to the sun than that of the earth ; 
while all the others being more distant from the sun than the 
earth, are called superior planets. The planets present great 
diversities among themselves in respect to distance from the 
sun, magnitude, time of revolution, and density. They differ 
also in regard to satellites, of which, as we have seen, the 
Earth and Neptune have each one, Jupiter has four, Saturn 
eight, and Uranus six ; while Mercury, Yenus, and Mars, have 

* Respecting the number of satellites belonging- to Uranus, there is some doubt, 
which will be considered under the history of that planet. 



176 THE PLANETS. 

none at all. It will aid the memory, and render our view of 
the, planetary system more clear and comprehensive, if we 
classify, as far as possible, the various particulars compre- 
hended under the foregoing heads. 

299. DISTANCES FROM THE SUN.* 

1. Mercury, 37,000,000 0.38709S1 

2. Yenus, 68,000,000 0.7233316 ' 

3. Earth, 95,000,000 1.0000000 

4. Mars, 145,000,000 1.5236923 

5. Planetoids, 250,000,000 2.6612SS5 

6. Jupiter, 495,000,000 5.2027760 

7. Saturn, 900,000,000 9.5387861 

8. Uranus, 1,800,000,000 19.1823900 

9. Neptune, 2,800,000,000 30.0318000 

The dimensions of the planetary system are seen from this 
table to be vast, comprehending a circular space nearly six 
thousand millions of miles in diameter. A railway car, trav- 
eling night and day at the rate of 20 miles an hour, and of 
course making 480 miles a day, would require about 50 days 
to travel round the Earth on a great circle, and about 500 days 
to reach the moon ; but it will give some idea of the vastness 
of the planetary spaces to reflect, that setting out from the 
sun, and traveling from planet to planet at the same rate, to 
reach Mercury would require about 200 years; Yenus, nearly 
400 ; the Earth, 542 ; Mars, more than 800 ; Jupiter, toward 
3,000 ; Saturn, above 5,000 ; Uranus, 10,000 ; Neptune, more 
than 16,000 ; and to cross the entire orbit of Neptune would 
require upward of 32,000 years. 

It may aid the memory to remark, that in regard to the 
planets nearest the sun, the distances increase in an arithmeti- 
cal ratio, while those most remote increase in a geometrical 
ratio. Thus, if we add 30 to the distance of Mercury, it gives 
us nearly that of Yenus ; 30 more gives that of the Earth ; 
while Saturn is nearly twice the distance of Jupiter, and 



* The distances in miles, as expressed in the first column, are to be treasured 
up in the memory, while the second column expresses the relative distance, that 
of the Earth being 1, from which a more exact determination may be made 
when required, the Earth's distance being taken at. 95,298,260 miles. 



DISTANCES- FROM THE SUN". . 177 

Uranus twice that of Saturn. If this, however, were a per- 
fectly correct rule, Neptune would be twice as far from the 
sun as Uranus, and therefore 3,600 millions of miles, whereas 
its actual distance is short of 3,000 millions. Between the 
orbits of Mars and Jupiter a great chasm appeared, which 
broke the continuity ; but the discovery of the planetoids has 
filled the void. A more exact law of the series is that called 
Bode's. law. It is as follows : if we represent the distance of 
Mercury by 4, and increase trie following terms by the prod- 
uct of 3. into the ascending powers of 2, 4 we shall obtain the 
relative distances of the planets from the sun. Thus, 

Mercury, . . 4 =4 

Venus, 4+3.2° = 7 

Earth, 4+3.2 1 = 10 

Mars, 4+3.2* = 16 

Planetoids, ...... 4+3.2 3 = 28 

Jupiter, 4+3.2 4 = 52 

Saturn, 4+3.2 5 =100 

Uranus, 4+3.2 6 =196 

Neptune, 4+3.2 7 =388 

Before the discovery of Neptune, Bode's law rudely ex- 
pressed the relative distances from the sun; but it signally 
fails of including the new planet, as it gives near 3,600 millions 
instead of the true distance, 2,800 millions. 

But the relative distances from the sun are accurately ob- 
tained by Kepler's third law, — that the squares of the periodic 
times are as the cubes of the distances (Art. 171). Thus the 
Earth's distance being previously ascertained by means of the 
sun's horizontal parallax (Art. 87), and the period of any other 
planet, as Jupiter, being learned from observation, we may say, 
as the square of the Earth's period (365.256 days) is to the 
square of Jupiter's period (4332.586 days), so is the cube of the 
Earth's distance to the cube of Jupiter's distance, the cube root 
of which will be the distance itself. Or, to express the same 
truth more concisely, 365.256 2 : 4332.S86 2 : : l 3 : 5.202 3 . Of 
course, this method can as yet be used only approximately for 
Neptune, which has described but a short portion of its orbit 
since its discovery. 

12 



178 THE PLANETS. 

300. MAGNITUDES. 

Diameter Mean apparent 

in Miles. Diameter. Mass. Volume. 

Mercury, 2,950* 8". 8 A 

Yenus, 7,800 17". 97 ft 

Earth, 7,912 100 1 

Mars, 4,500 6". 15 i 

Planetoids, .... 0".5 uncertain. 

Jupiter, .... 89,000 37". 37,171 1,400 

Saturn, ..... 79,000 16". 11,121 1,000* 

Uranus, .... 35,000 4". 1,564 86 

Neptune, .... 31,000* 2".5 672 60 

Diagrams and orreries, as usually constructed, wholly fail of 
giving any just conceptions of the distances of the planets 
from the sun and from each other. If we represent, for in- 
stance, the distance of the earth by 1 foot, we shall require 30 
feet in order to reach the place of Neptune; and when we 
have constructed a diagram on so large a scale, we must still 
bear in mind that each foot represents a space of nearly 100 
millions of miles.f 

We remark here a great diversity in regard to magnitude — 
a diversity which does not appear to be subject to any definite 
law. While Yenus, an inferior planet, is nine-tenths as large 
as the earth, Mars, a superior planet, is only one-sixth, while 
Jupiter is fourteen hundred times as large. Although several 
of the planets, when nearest to us, appear brilliant and large 
when compared with the fixed stars, yet the angle which they 
subtend is very small, that of Yenus, the greatest of all, never 
exceeding about 1', or more exactly 61". 2, and that of Jupiter, 
when greatest, being only about f of a minute. 

* Hind. 

f For the purposes of illustration to a class or to a popular audience, the fol- 
lowing plan of representation is recommended, not only for the entire solar sys- 
tem, but for each of the subordinate systems, as that of Jupiter or Saturn. 
Stretch upon the wall a piece of black cambric, as long as the room will allow, 
say 30 ft. This length may be taken for the radius of Neptune's orbit. At one 
end, attach a circle of white cloth, $ inch in diameter for the sun. From it as 
a center describe arcs across the cloth at proper distances for the several orbits, 
and sew white tape on these arcs. We then have the planetary distances, and 
the size of the sun, upon one scale. The planets themselves can not be repre- 
sented, since the largest of them would be almost invisibly small. The cloth 
may be conveniently rolled up, when not in use. 



INFERIOR PLANETS — MERCURY AND VENUS. 179 

Tlie distance of one of the near planets, as Venus or Mars, 
may be determined from its parallax ; and the distance being 
known, its real diameter can be estimated from its apparent 
diameter, in the same manner as we estimate the diameter of 
the sun. (Art 145.) 





301. 


PERIODIC TIMES. 






Sidereal revolution. 




Mean daily motion. 


Mercury, 


3 months. 


, or 


88 days, 


4° 5' 32".6 


Venus, 


n " 


a 


224: " 


1° 36' 7".8 


Earth, 


1 year, 


a 


365 " 


0° 59' 8".3 


Mars, 


2 " 


u 


687 " 


0° 31' 26".7 


Ceres, 


H " 


u 


1,681 " 


0° 12' 50".9 


Jupiter, 


12 " 


a 


4,332 « 


0° 4' 59".3 


Saturn, 


29 " 


u 


10,759 " 


0° 2' 0".6 


Uranus, 


84: " 


u 


30,686 " 


0° 0'42".4 


Neptune, 164+ " 


a 


60,127 " 


0° 0' 21".5 



From this view it appears that the planets nearest the sun 
move most rapidly. Thus Mercury performs nearly 350 revo- 
lutions while Uranus performs one. This is evidently not 
owing merely to the greater dimensions of the erbit of Uranus, 
for the length of its orbit is not 50 times that of the orbit of 
Mercury, while the time employed in describing it is 350 
times that of Mercury. Indeed, this ought to follow from 
Kepler's law, that the squares of the periodic times are as the 
cubes of the distances ; from which it is manifest that the times 
of revolution increase faster than the dimensions of the orbit. 
Accordingly, the apparent progress of the most distant planets 
is exceedingly slow, the rate of Uranus being only 42 ;/ .4 per 
day ; so that for weeks and months, and even years, this planet 
but slightly changes its place among the stars. 

The planets are divided into two classes : first, the inferior, 
which have their orbits nearer to the sun than that of the 
earth ; and secondly, the superior, which have their orbits ex- 
terior to the earth's orbit. 

THE INFERIOR PLANETS, MERCURY AND VENUS. 

302. The inferior planets, Mercury and Venus, having their 
orbits far within that of the earth, appear to us as attendants 



180 



THE PLANETS. 



upon the sun. Mercury never appears further from the sun 
than 29° (28° 48'), and seldom so far ; and Yenus never more 
than about 47° (47° 12'). Both planets, therefore, appear either 
in the west soon after sunset, or in the east a little before sun- 
rise. In high latitudes, where the twilight is prolonged, Mer- 
cury can seldom be seen with the naked eye, and then only at 
the periods of its greatest elongation.* The reason of this will 
readily appear from the following diagram. 

Let S represent the sun, E the earth, and MK Mercury at 
its greatest elongations from the sun, and OZP a portion of the 
sky. Then, since we refer all distant bodies to the same con- 
cave sphere of the heavens, we should see the sun at Z and 
Mercury at O, when at its greatest eastern elongation, and at P 
when at its greatest western elongation ; and while passing 
from M to 1ST through Q, it would appear to describe the arc 
OP ; and while passing from N to M, through R, it would 
appear to run back across the sun on the same arc. It is 
further evident that it would be visible only when at or near 
one of its greatest elongations ; being at all other times so near 
the sun as to be lost in his light. 




303. A planet is said to be in conjunction with the sun 
when it is seen in the same part of the heavens with the sun, 



* Copernicus is said to have lamented, on his death-bed, that he had never 
been able to obtain a sight of Mercury ; and Delambre, a great French astrono- 
mer, sa-v^ it but twice. 



INFERIOR PLANETS MERCURY AND VENUS. 181 

or when it has the same longitude. Mercury and Yenus have 
each two conjunctions, the inferior and the superior. The 
inferior conjunction is its position when in conjunction on the 
same side of the sun with the earth, as at Q in the figure ; the 
superior conjunction is its position when on the side of the sun 
most distant from the earth, as at R. 

304. The period occupied by a planet between successive 
conjunctions of the same kind is called its synodical revolution. 
Both the planet and the earth being in motion, the time of the 
synodical revolution exceeds that of the sidereal revolution of 
Mercury or Yenus ; for when the planet comes round to the 
place where it before overtook the earth, it does not find the 
earth at that point, but far in advance of it. Thus, let Mer- 
cury come into its inferior conjunction at Q (Fig. 60). In 
about 88 days the planet will come round to the same point 
again ; but meanwhile the earth has moved forward through 
nearly a fourth part of her revolution, and will continue to 
move onward while Mercury, with a swifter motion, is follow- 
ing on to overtake her, the case being analogous to the hour 
and minute hand of a clock. Having the sidereal period of a 
planet, which may always be accurately determined by obser- 
vation, we may ascertain its synodical period, as follows : 

By the table in article 301, the mean daily motion of Mer- 
cury is 4° 5' 32".6 = 14732".6, and that of the earth is 59' 
8".3=354:8".3. Therefore, 14r732 / '.6-3548.3=11184 // .3, which 
is the average gain of Mercury over the earth in a day. But 
in order to overtake the earth, Mercury must complete one 
revolution and as much of another as the earth has performed, 
nntil the planet overtakes it ; that is, the planet must gain an 
entire revolution. Now, 

11184".3 : 1 day : : 360° : 115,8 days, the synodical period of 
Mercury. In like manner, the daily gain of Yenus is 2219".5, 
and 

2219".5 : 1 day : : 360° : 583.9 days, the synodical period of 
Yenus. 

•305. The motion of an inferior planet is direct in passing 
through its superior conjunction, and retrograde in passing 
through its inferior conjunction. 



182 



THE PLANETS. 



Thus Yenus, while going from B through D to A (Fig. 61), 
moves in the order of the signs, or from west to east, and 
would appear to traverse the celestial vault B' S' A' from right 
to left ; but in passing from A through C to B, her course 
would be retrograde, returning on the same arc from left to 
right. If the earth were at rest, therefore (and the sun, of 
course, at rest), the inferior planets would appear to oscillate 
backward and forward across the sun. But it must be recol- 
lected that the earth is moving in the same direction with the 
planet, as respects the signs, but with a slower motion. This 
modifies the apparent motions of the planet, accelerating it in 
the superior, and retarding it in the inferior conjunctions. 
Thus, in figure 61, Yenus, while moving through BDA, would 

Fig. 61. 




seem to move in the heavens from B f to A', were the earth at 
rest ; but meanwhile the earth changes its position from E to 
E', by which means the planet is not seen at A', but at A", 
being accelerated by the arc A 'A", in consequence of the 
earth's motion. On the other hand, when the planet is pass- 
ing through its inferior conjunction ACB, it would appear to 
move backward in the heavens A' to B', if the earth were at 
rest, but from A' to B", if the earth has, in the mean time, 
moved from E to E', being retarded by the arc B'B". Al- 
though the motions of the earth have the effect to accelerate 



INFERIOR PLANETS MERCURY AND VENUS. 183 

the planet in the superior, and to retard it in the inferior con- 
junction, jet on account of the greater distance, the apparent 
motion of the planet is much slower in the superior than in 
the inferior conjunction. 

306. In passing from either conjunction to the other, an 
inferior planet is stationary at a point a little way from the 
greatest elongation toward the inferior conjunction. 

If the earth, were at rest the stationary points would be at 
the greatest elongations A and B ; for then the planet would 
be moving directly toward or from the earth, and would be 
seen for some time in the same place in the heavens ; but the 
earth itself is moving nearly at right angles to the line of the 
planet's motion, and therefore a direct apparent motion is 
given to the planet. Hence we need to choose such a position 
for the planet that its retrograde movement shall be just suffi- 
cient to counteract this. Of course it must be on the arc ACB. 
But, as the planet's angular velocity is much greater than the 
earth's, it must be near A or B, where the motion is quite ob- 
lique to our own, else the retrogradation will be too rapid to 
neutralize the direct motion caused by the earth's progress. 
The stationary point for Mercury is at an elongation of 15° or 
20° from the sun, that of Yenus at about 29°. 

307. Mercury and Yenus exhibit to the telescope phases 
similar to. those of the moon. 

When on the side of their inferior conjunction, as from A to 
B through C (Fig. 61), these planets appear horned, like the 
moon in her first and last quarters ; and when on the side of 
their superior conjunctions, as from B to A through D, they 
appear gibbous. At the moment of superior conjunction, the 
whole enlightened orb of the planet is turned toward the earth, 
and the appearance would be that of the full moon, but the 
planet is too near the sun to be commonly visible. All these 
changes of figure, resulting from the different positions of the 
planet with respect to the sun and earth, will be readily un- 
derstood by inspecting the diagram (Fig. 61). 

• The phases show that these bodies are not self-luminous, but 
shine only as they reflect to us the light of the sun ; and all 
the planets in some way give evidence of the same fact. 



184: 



THE PLANETS. 



Fie:. 62. 



308. The distance of an inferior planet from the sun, may 
he found by observations at the time of its greatest elongation. 

Thus if E (Fig. 62) be the place of the earth, and C that of 
Venus at the time of her greatest elongation, the angle SCE 
will be known, being a right angle. The angle SEC is the 
greatest elongation ; this is known by observation. Hence, if 
ES is considered to be known, Ead : sin E : : SE : SC, which 
is the distance of the planet from the sun. But 
if SE be not definitely known, then this pro- 
portion gives only the ratio of the distances of 
the earth and the inferior planet from the sun. 
In finding the earth's distance from the sun by 
means of the transit of Yenus (Art. 318), this 
ratio will be employed. If the orbits were 
both circles, this method would be very exact ; 
but being elliptical, we obtain the mean value 
of the radius SC by observing its greatest elon- 
gation in different parts of its orbit.* 

309. The orbit of Mercury is more eccentric, 
and more inclined to the ecliptic, than that of E 

any other planet /f while that of Venus is nearly circular, and 
but little inclined to the ecliptic. 

The eccentricity of the orbit of Mercury is nearly J of its 
semi-major axis, while that of Yenus is T Js ; and that of the 
earth only -§q ; the inclination of Mercury's orbit is 7°, while 
that of Yenus is only 3i°.$ At the perihelion, Mercury is 
only 29 millions of miles from the sun, while at the aphelion 
his distance is 44 millions, a variation of 15 millions, and more 
than five times as great as in the case of the earth. On ac- 
count of his different distances from the earth, Mercury is also 
subject to much variation in his apparent diameter, which is 
12" in perigee, but only 5" in apogee. 

310. After the mean distance has been found (Art. 307), 
the periodic time is obtained, by applying Kepler's third law 
to the orbit of the planet, and that of the earth. From this is 
calculated the synodical period (Art. 304). 




° Herschel's Outlines, p. 275. 
% Baily's Tables. 



f The asteroids are of course excepted. 



INFERIOR PLANETS MERCURY AND VENUS. 185 

The synodical period is also obtained very accurately from 
the interval between two transits across the sun's disk. 

311. An inferior planet is brightest at a certain point be- 
tween its greatest elongation and inferior conjunction. 

As to distance, the planet would give us most light when 
nearest, i. e., at the inferior conjunction ; but so far as the 
phase is concerned, it would give us most at the superior con- 
junction, where the planet is at the full. Of course, the maxi- 
mum is at some point between the two conjunctions; and by 
calculation it is found between the inferior conjunction and 
greatest elongation, within a few degrees of the latter. Yenus 
appears most luminous when about 40° from the sun, and is 
sometimes visible all day. 

312. Mercury and Venus both revolve on their axes in 
nearly the same time with the earth. 

The diurnal period of Mercury is a little greater than that of 
the earth, being 24h. 5m. 28s., and that of Venus is a little 
less than the earth's, being 23h. 21m. 7s. The revolutions on 
their axes have been determined by means of some spot or 
mark seen by the telescope, as the revolution of the sun on his 
axis is ascertained by means of his spots. 

313. Yenus is regarded as the most beautiful of the plan- 
ets, and is well known as the morning and evening star. The 
most ancient nations did not indeed recognize the evening and 
morning star as one and the same body, but supposed they 
were different planets, and accordingly gave them different 
names, calling the morning star Lucifer, and the evening star 
Hesperus. At her period of greatest splendor, Yenus casts a 
shadow, and is sometimes visible in broad daylight. This oc- 
curred in a very striking manner in September, 1852, Yenus 
being on the meridian about 9 o'clock, a. m., and her northern 
declination nearly 15 degrees. Although not 15° from the in- 
ferior conjunction, and consequently exposing only a portion 
of her disk, like that of the moon when three or four days old, 
yet her light is then estimated as equal to that of twenty stars 
of the first magnitude.* At her period of greatest elongation, 

* Francoeur, Uranogrophy, p. 125. 



186 THE PLANETS. 

Yenus is visible from three to four hours after the setting, or 
before the rising of the sun. 

314. Every eight years, Venus forms her conjunctions with 
the sun in the sam,e part of the heavens. 

The sidereal period of Yenus being 224.7 days, and that of 
the earth 365.256 days, thirteen revolutions of Yenus are ac- 
complished in nearly the same time as eight revolutions of the 
earth: for 224.7x13=2921, and 365.256x8=2922, At the 
end, therefore, of 2922 days, or eight years, the two bodies will 
come round to the same point of the heavens, and be in the 
same situation in their respective orbits, as at the beginning. 
Consequently, whatever appearances of this planet arise from 
its positions with respect to the earth and the sun (as, for ex- 
ample, being visible in the daytime), they are repeated every 
eight years in nearly the same form. 



TRANSITS OF THE INTERIOR PLANETS. 

315. The transit of Mercury or Venus, is its passage across 
the sun's dish, as the moon passes over it in a solar eclipse. 

As a transit takes place only when the planet is in inferior 
conjunction, at which time her motion is retrograde (Art. 305), 
it is always from left to right, and the planet is seen projected 
on the solar disk in a black round spot. Were the orbits of 
these planets coincident with the earth's orbit, a transit would 
occur at some part of the earth at every inferior conjunction, 
as there would be an eclipse of the sun at every new moon, 
were the moon's revolution in the plane of the ecliptic. But 
the orbit of Yenus makes an angle of 3|-° with that of the 
earth, and the orbit of Mercury an angle of 7° ; and, more- 
over, the apparent diameter of each of these bodies is very 
small, both of which circumstances conspire to render a transit 
a comparatively rare occurrence, since it can happen only 
when the sun, at the time of an inferior conjunction, happens 
to be at, or extremely near the planet's node. The nodes of 
Mercury lie in that part of the earth's orbit which it passes in 
the months of May and November. It is only in these months, 
therefore, that transits of Mercury can occur. For a similar 



TRANSITS OF THE INFERIOR PLANETS. 187 

reason, those of Terms occur only in June and December. 
Since the nodes of both planets have a small retrograde mo- 
tion, the months in which transits occur, will change in the 
course of ages ; but the months for transits will for a long 
time remain the same as at present, since the nodes of Mer- 
cury change their places only 13', and those of Yenus only 31' 
in a century.* 

The first prediction of this phenomenon was made by Kepler, 
and was that of a transit of Mercury, which occurred on the 
7th of November, 1631. As early as 1629, Kepler announced 
to astronomers that his tables gave the latitude of Mercury, at 
the conjunction which was to take place on that day, less than 
the sun's semi-diameter ; consequently, that the planet, in 
passing by the sun, would be nearer the sun's center than the 
length of the sun's radius, and of course appear on his disk. 
The event corresponded to the prediction. The transit of 
Mercury, which occurred on the 8th of November, 1848, was 
the 25th since the one predicted by Kepler, averaging nearly 
one in 8 years, although they take place at very unequal in- 
tervals. 

316. The shortest interval between two successive transits 
of Mercury is 3-j- years, and of Venus, 8 years: but sometimes 
they are separated by long intervals, especially those of Yenus. 
The only transits of Yenus in the 19th century, are in 1874 
and 1882; and in the 20th, not a single one will occur; the 
intervals being 8, 121i 8, 105^, 8, 1211, & c ., years. The in- 
tervals between the transits of Mercury, from 1848, through 
the century, are 13, 7, 9i, 3J, 9|, and 3J years. The shortest 
interval for Mercury at the same node is 7 years ; hence, at 
opposite nodes, two transits may occur within 3^ years. More 
of the transits of Mercury happen in November than in May, 
because the orbit of this planet (w T hich has a great eccentricity, 
Art. 308), is so situated that in November the planet is near its 
perihelion, and is then more likely to be projected on the sun, 
in passing its inferior conjunction, than in a part of its orbit 
more distant from the sun. 
•Let us see how the intervals between the transits of Mercury 

« Hind. 



188 THE PLANETS. 

or Venus are found. Since Venus, for example, completes one 
revolution around the sun in 224.7 days, and the earth in 
365.256, and since the number of times each will revolve in a 
given period is inversely as the time of one revolution, there- 
fore, in 224,700 revolutions of the earth, and 365,256 revolu- 
tions of Yenus, the two bodies would meet exactly at the same 
node as before. But 224,700 : 365,256 : : 8 : 13 nearly ; so that 
transits of Yenus are sometimes repeated at intervals of 8 years, 
and if the ratio of 8 to 13 were exactly that of the two first 
terms of the proportion, we should have a transit of Yenus 
every 8 years. The ratio of 227 to 369 is still nearer that of 
those terms ; and hence a transit after 227 years is still more 
probable ; but since there are two nodes, the chance is doubled, 
so that a transit is highly probable after an interval of 113J 
years. The two transits of Yenus in the 18th century occurred 
in June, 1761, and June, 1769, 8 years apart; the two transits 
of the 19th century are in December, 1874, and December, 
1882, the intervals being 105^ and 8 years. The average long 
interval is 113^ years, but it may be lengthened to 121J, or 
shortened to 105J, according as the preceding transit took place 
before or after passing the node. 

317. The great interest attached by astronomers to a transit 
of Yenus, arises from its furnishing the most accurate means 
in our power of determining the surfs horizontal parallax- — an 
element of great importance, since it leads to a knowledge of 
the distance of the earth from the sun, and consequently, by 
the application of Kepler's third law (Art. 183), of the dis- 
tances of all the other planets. Hence, in 1769, great efforts 
were made throughout the civilized world, under the patron- 
age of different governments, to observe this phenomenon 
under circumstances the most favorable for determining the 
parallax of the sun. The method of finding the parallax of a 
heavenly body, described in Art. 85, can not be relied on to a 
greater degree of accuracy than 4". In the case of the moon, 
whose greatest parallax amounts to about 1°, this deviation 
from absolute accuracy is not material, but it amounts to 
nearly half the entire parallax of the sun ; and since the dis- 
tance is inversely as the horizontal parallax, such an error 
would make the distance of the sun either twice as great, or 



TRANSITS OF THE INFERIOR PLANETS. 189 

only two-thirds as great as the true distance, according as the 
parallax was 4" below or 4" above the truth. 

318. If the sun and Yenus were equally distant from us, 
they would be equally affected by parallax as viewed by spec- 
tators in different parts of the earth, and consequently their 
relative situation would not be altered by such a difference in 
the points of view ; but since Yenus, at the inferior conjunc- 
tion, is only about one-third as far off as the sun, her parallax 
is proportionally greater, and therefore spectators, at distant 
points, will see Yenus projected on different parts of the solar 
disk ; and as the planet traverses the disk, she will appear to 
describe chords of different lengths, by means of which the 
duration of the transit may be estimated at different places. 
The difference in the duration of the transit, as viewed from 
opposite parts of the earth, does not amount to many minutes ; 
but to make it as large as possible, places very distant from 
each other are selected for observation. Thus, in the transit 
of 1769, among the places selected, two of the most favorable 
were "Wardhus, in Lapland, and Otaheite (now written Tahiti), 
one of the Society Islands, in the South Pacific Ocean, to which 
place the celebrated Captain Cook was dispatched by the 
British government for the express purpose of observing the 
transit. 

Although the exact determination of the sun's horizontal 
parallax by this method is a very complicated and difficult 
problem, yet the principle on which the process depends ad- 
mits of an easy illustration. Let E (Fig. 63) be the earth, Y 
Yenus, and S the center of the sun. Suppose A and B two 



Fig. 63. 




B 

3E 



observers at the extremities of that diameter of the earth which 
is perpendicular to the orbit of Yenus. At a certain moment 
the spectator A will see Yenus on the sun's disk at a, and the 
spectator B will see it at b ; and since AY and BY may be 
considered as equal to each other, as also Yb and Ya ; there- 



190 THE PLANETS. 

fore the triangles YAB and Yah are similar ; and AY : aV : : 
AB : ah. But AB is known ; also the ratio of AY : aV (Art. 
308) ; hence ah, the distance between the points at which the two 
observers see Y projected on the sun's disk, is known in miles. 
We need now to obtain the angular value of ah. The observers 
carefully note the instant when Yenus touches the disk, at the 
beginning of the transit, and also at the end. Thus the time 
of making the transit, as seen by each observer, is accurately 
obtained. But since the angular motion per hour, both of the 
planet and of the sun, is known, this time of crossing the disk 
can be changed into an arc ; and we thus have the number of 
minutes of a degree in the chord cd, and also the number in ef y 
and of course in their halves ca and eh. But cS and eS, the 
angular semi-diameter of the sun at the same time is known ; 
hence, in the right-angled triangles cSa, eSb, we readily find 
the minutes in S#, S5, the difference between which is the 
angular value of ah. We have, therefore, ascertained what 
angle is subtended by a line of given length, when placed at 
the sun and viewed from the earth ; or, which is the same 
thing, placed at the earth and viewed from the sun ; and 
therefore we know what angle at the sun is subtended by the 
earth's semi-diameter, which is the sun's horizontal parallax. 

The observers can not, probably, be at points diametrically 
opposite, nor can they remain stationary during the transit, on 
account of diurnal motion ; therefore allowance must be made 
for these circumstances. The line ah will be more accurately 
measured, according as the transit occurs nearer to the edge of 
the sun's disk, because of the greater inequality in the length 
of the chords. The solar parallax is so small that several sta- 
tions should be occupied, so as to obtain a number of inde- 
pendent results. The parallax of the sun, as measured in 1769, 
is 8".5776 * 

Yenus when on the side of her inferior conjunction, and 
Mars when near his opposition, each comes comparatively near 
to the earth, and at these times exhibits a large horizontal 
parallax. That of Yenus, especially, may be obtained with 
great accuracy when she is near her greatest elongation ; and 



* Delambre, t. ii ; Vince, Complete Syst., vol. i. ; Woodhouse, p. 754; Her- 
schel's Outlines, p. 255. 



TRANSITS OF THE INFERIOR PLANETS. 191 

since it is easy, by article 308, to determine, at that time, the 
ratio of her distance from the sun to the earth's distance, it is 
a matter of great interest to astronomy to have the parallax of 
Yenns, when thus situated, accurately found. For this pur- 
pose, the government of the United States, in 1849, sent an 
expedition, under Lieutenant Gilliss, to Chili, in order to take 
observations on Mars and Yenus, especially the latter, during 
1850, 1851, and 1852, in concert with the Observatory at 
"Washington. These observations seem to indicate that the 
solar parallax is a little less than stated above, probably near- 
er 8".5 than 8".6.* 

319. During the transits of Yenus over the sun's disk in 
1761 and 1769, a sort of penumbral light was observed around 
the planet by several astronomers, which was "thought to indi- 
cate an atmosphere. This appearance was particularly observ- 
able while the planet was coming on and going off the solar 
disk. The total immersion and emersion were not instanta- 
neous ; but as two drops of water when about to separate form 
a ligament between them, so there was a dark shade stretched 
out between Yenus and the sun, and when the ligament broke, 
the planet seemed to have got about an eighth part of her 
diameter from the limb of the sun.f The presence of an at- 
mosphere is also indicated by appearances of twilight and 
indications of a horizontal refraction.^ 

Although no satellite has hitherto been discovered attending 
either Mercury or Yenus, yet suspicions have, at different 
times, been entertained of a satellite belonging to Yenus. 
None has been seen in any of the transits of Yenus ; and al- 
though the distance of the satellite (if one exists) from the 
primary might have been too great to be projected with the 
primary on the sun, yet its absence on each of these occasions 
has strengthened the belief of astronomers that no such satel- 
lite exists. 

• See Astron. Journ., Oct., 1858. f Edin. Encyc, Art. Astronomy. % Hind. 



CHAPTER X. 

OF THE STTPEEIOE PLANETS — MAES, THE PLANETOIDS, JUPITEE, 
SATTJEN, TJEANUS, AND NEPTTJNE. 

320. The Superior planets are distinguished from the In- 
ferior, by being seen at all distances from the sun from 0° to 
180°. Having their orbits exterior to that of the earth, they 
of course never come between us and the sun, that is, they 
have never any inferior conjunction like Mercury and Venus, 
but they are seen in superior conjunction and in opposition. 
Nor do they, like the inferior planets, exhibit to the telescope 
different phases; but, with a single exception, they always 
present the side that is turned toward the earth fully enlight- 
ened. This is owing to their great distance from the earth : 
for were the spectator to stand upon the sun, he would, of 
course, always have the illuminated side of each of the planets 
turned toward him ; but so distant are all the superior plan- 
ets except Mars, that they are viewed by us very nearly as 
they would be if we actually stood on the sun. 

321. Maes is a small planet, his diameter being only about 
half that of the earth, or 4500 miles.* He also, at times, comes 
nearer to us than any other planet except Venus. His mean 
distance is 145,200,000 miles; but in consequence of the eccen- 
tricity of his orbit, the distance varies greatly, the ' difference 
between the perihelion and aphelion distances being 27,000,000 
miles. Mars is always near the ecliptic, never varying from it 
2°. He is distinguished from all the other planets by his deep 
red color and fiery aspect ; but his brightness and apparent 
magnitude vary much at different times, being sometimes 
nearer to us than at others by the whole diameter of the 
earth's orbit, that is, by about 190,000,000 miles. When Mars 
is on the same side of the sun with the earth, or at his opposi- 
tion, he comes within 50,000,000 miles of the earth, and, rising 

» Hind. 



MAES. 193 



about the time the sun sets, surprises us by his magnitude and 
splendor ; but when he passes to the other side of the sun to 
his superior conjunction, he dwindles to the appearance of a 
small star, being then 240,000,000 miles from us. Thus, let M 




(Fig. 64) represent Mars in opposition, and M' in superior 
conjunction, it is obvious that the planet must be nearer to us 
in the former situation than in the latter by the whole diame- 
ter of the earth's orbit. 

322. Mars is the only one of the superior planets which 
exhibits phases. "When he is toward the quadratures at Q or 
Q', it is evident from the figure that only a part of the circle 
of illumination is turned toward the earth, such a portion of 
the remoter part of it being concealed from our view as to 
render the form more or less gibbous. 

323. When viewed with a powerful telescope, the surface 
of Mars appears diversified with numerous varieties of light 
and shade. The region around the poles is marked by white 
spots, which vary their appearance with the changes of the 
seasons in the planet. Hence Dr. Herschel conjectured that 
they are owing to ice or snow which occasionally accumulates 
and melts, according to the position of each pole with respect 

13 



194- THE PLANETS. 

to the sun.* It has been common to ascribe the ruddy light 
of this planet to an extensive and dense atmosphere, which was 
supposed to be distinctly indicated by the gradual diminution 
of light observed in a star as it approached very near to the 
planet in undergoing an occultation ; but more recent observa- 
tions afford no such evidence of an atmosphere.f By observa- 
tions on the spots, we learn that Mars revolves on his axis in 
very nearly the same time with the earth (24h, 39m. 21 s .3); 
and that the angle between his equator and the plane of his 
orbit is also nearly the same as between the earth's equator r.nd 
the ecliptic, the former being 28° 42', the latter 23° 28', so that 
the changes of seasons on Mars must resemble our own. 

No satellite has ever been discovered belonging to Mars, 
although being situated at a greater distance from the sun than 
our globe, it might seem more especially to need such a 
luminary to cheer its dark nights. As the diurnal rotation of 
Mars is performed in nearly the same time as the earth, we 
should expect a similar flattening of the poles. Such is the 
fact, and the ellipticity of Mars exceeds that of the earth, being 
about one-fiftieth,:): while the earth's ellipticity is one three- 
hundreth. This difference in the conjugate diameters may be 
readily observed when the planet is in opposition, the whole 
enlightened disk being then presented to us. 

324. Mars being comparatively near to us when on the 
same side of the sun with the earth, and the ratio of his dis- 
tance from the sun to that of the earth being easily obtained, 
astronomers have sought by means of his parallax, as by that 
of Venus, to find the sun's horizontal parallax. But the meth- 
od by observations on Yenus, as described in Art. 318, is more 
to be relied on. 

325. The Asteroids or Planetoids compose a group of very 
small planets, indefinite in number, whose orbits lie beyond 
that of Mars, at the distance of about 250,000,000 miles from 
the sun. The discovery of them commenced with the begin- 
ning of the present century. Kepler had long before noticed 
a large interval between Mars and Jupiter, which seemed to 

* Phil. Trans., 1784. | Sir James South, Phil. Trans., 1833. % Hind. 



NEW PLANETS, OR ASTEROIDS. 195 

break the continuity of the series ; and about the close of the 
last century, Bode, of Berlin, showed that a series of numbers 
following a certain law (Art. 299) would express quite accu- 
rately the distances from the sun, even to Uranus, which had 
just been discovered, if only the vacancy between Mars and 
Jupiter were supplied. So strongly were astronomers impressed 
with the idea that a planet would be found between Mars and 
Jupiter, that, in hope of discovering it, an association was 
formed on the continent of Europe of twenty-four observers, 
who divided the sky into as many zones, one of which was 
allotted to each member of the association. The discovery of 
the first of these bodies was, however, made accidentally by 
Piazza, an astronomer of Palermo, on the 1st of January, 1801. 
It was shortly afterward lost sight of, on account of its prox- 
imity to the sun, and was not seen again until the close of the 
year, when it was rediscovered in Germany. Piazza called it 
Ceres, in honor of the tutelary goddess of Sicily, and her em- 
blem, the sickle ? , has been adopted as the appropriate sym- 
bol. The difficulty of finding Ceres induced Dr. Olbers, of 
Bremen, to examine, with particular care, all the small stars 
that lie near her path, as seen from the earth ; and, while pros- 
ecuting these observations, in March, 1802, he discovered 
another similar body, very nearly at the same distance from 
the sun, and resembling the former in many other particulars. 
The discoverer gave to this second planet the name of Pallas, 
choosing- for its symbol the lance $, the characteristic of 
Minerva. 

326. The most surprising circumstance connected with the 
discovery of Pallas, was the existence of two planets at nearly 
the same distance from the sun, and apparently having a com- 
mon node ; a circumstance that indicated an identity of origin. 
On account of this singularity, Dr. Olbers was led to conjecture 
that Ceres and Pallas are only fragments of a larger planet 
which had formerly circulated around the sun at this distance, 
and been shattered by some great convulsion. 

In 1804, near one of the nodes of Ceres and Pallas, a third 
planet was discovered. This was named Juno, and the charac- 
ter § was adopted for its' symbol, representing the starry scep- 
ter of the goddess. In 1807, a fourth planet, Vesta, was dis- 



1&6 THE PLANETS. 

covered, and for its symbol the character fi was chosen — an 
altar surmounted with a censer holding the sacred fire. It is 
one of the largest of the asteroids, and has sometimes been seen 
by the naked eye. 

327. From 1807 to 1845, a period of nearly forty years, no 
more of these small planets were discovered, and, up to this 
time, by the asteroids were meant the fonr little planets already 
enumerated — Ceres, Pallas, Juno, and Yesta. Meanwhile, 
very accurate maps of the stars, including all up to the tenth 
magnitude, had been published, especially in the region of the 
zodiac, and astronomers scrutinized these with such extreme 
closeness, that any wanderer appearing among them was likely 
to be immediately detected. Since 1845, new ones have been 
added to the list nearly every year. At the beginning of 1850, 
the whole number known was 10 ; of 1855, 23 ; of 1860, 57. 
Though feminine mythological names have been applied to 
nearly all of them, yet they are better designated by a small 
circle inclosing a number which expresses the order of their 
discovery : thus Ceres is Q ; Thetis, (*) ; Pandora, 0, &c. 

They vary somewhat in their mean distances, and of course 
in their periods. Some of them have orbits more eccentric, 
and others more inclined to the ecliptic, than any of the larger 
planets. They are too small to be measured with any certainty ; 
the largest, Ceres, is not estimated to be more than 160 miles in 
diameter. It is probable that all of them united would form 
but an inconsiderable planet. Some of them are attended by a 
nebulosity, which indicates that they have an extensive atmos- 
phere. The different planetoids have been first discovered by 
observers of many countries — England, Prance, Germany, Italy, 
and America. 

328. Jupiter is distinguished from all the other planets by 
his great magnitude. His diameter is 89,000 miles, being more 
than 11 times, and his volume more than 1400 times that of the 
earth. His figure is strikingly spheroidal, the equatorial ex- 
ceeding the polar diameter in the ratio of 107 to 100,* which 
is 21 times as great as the earth's ellipticity. This flattening 

* Herschel. 



JUPITER. 197 

of the poles is indeed quite perceptible by the telescope, and 
is obvious to the eye in a correct drawing of the planet. (See 
Frontispiece.) Such a figure might naturally be expected from 
the rapidity of his diurnal revolution, which is accomplished 
in about 10 hours (9k. 55m. 21 s .3).* 

A place on tke equator of Jupiter must revolve 450 miles 
per minute, or 27 times as fast as a place on the terrestrial 
equator. The distance of Jupiter from the sun is 495,000,000 
miles (495,817,000).f His plane of rotation is but slightly in- 
clined to the plane of his orbit (only about 3°), and consequent- 
ly his climate experiences but a slight change of seasons. 

329. The view of Jupiter through a good telescope is one 
of the most magnificent and interesting spectacles among the 
heavenly bodies. The disk expands into a large and bright 
orb like the full moon ; the spheroidal figure which theory as- 
signs to revolving worlds is here palpably exhibited to the eye; 
across the disk, arranged in parallel stripes, are discerned sev- 
eral dusky bands, called belts; and four bright satellites, always 
In attendance, but ever varying their positions, compose a 
splendid retinue. Indeed, astronomers gaze with peculiar in- 
terest on Jupiter and his moons, as affording a miniature rep- 
resentation of the whole solar system ; repeating, on a smaller 
scale, the same revolutions, and exemplifying, in a manner 
more within the compass of our observation, the same laws as 
regulate the entire assemblage of sun and planets. 

330. The -Belts of Jupiter are variable in their number and 
dimensions. With smaller telescopes only one or two are seen 
across the equatorial regions ; but with more powerful instru- 
ments the number is increased, covering a great part of the 
disk. Occasionally these belts retain nearly the same form and 
positions for many months together, while at other times they 
undergo great and sudden changes, and in one or two instances, 
they have been observed to break up and spread themselves 
over the whole face of the planet. The prevailing opinion 
among astronomers in reference to the nature of these belts is, 
that they are produced by disturbances in the planet's atmos- 

~ Airy. f Hind. 



198 THE PLANETS. 

pliere, which occasionally render its dark body visible ; and, 
as the belts are found to traverse the disk in lines uniformly 
parallel to Jupiter's equator, they are inferred to be connected 
with the rotation of the planet upon its axis, the great rapidity 
of which would naturally produce peculiarities in its atmos- 
pheric phenomena. 

331. The Satellites of Jupiter may be seen with a telescope 
of very moderate powers. Even a common spy-glass will 
enable us to discern them. Indeed, being nearly equal in bril- 
liancy to the smallest stars visible to the naked eye, a slight 
increase of optical power brings them into view ; and some few 
persons, endowed with extraordinary powers of vision, have 
supposed that they saw one of these little bodies without the 
aid of instruments ; but on applying the telescope it has been 
found that three of the satellites have approached so near to- 
gether as to appear like one.* In the largest telescopes, they 
severally appear as bright as Sirius does to the naked eye. 
With such an instrument, the view of Jupiter with his moons 
and belts is truly a magnificent spectacle — a world within it- 
self. As the orbits of the satellites do not deviate far from the 
plane of the ecliptic, and but little from the equator of the 
planet (which nearly coincides with the ecliptic), they are 
usually seen almost in a straight line extending across the cen- 
tral part of the disk. (See Frontispiece.) 

332. Jupiter's satellites are distinguished from one another 
by the denominations of first, second, third, and fourth, accord- 
ing to their relative distances from the primary, the first being 
that which is nearest to him.f Their apparent motion is os- 
cillatory, like that of a pendulum, going alternately from their 
greatest elongation on one side to their greatest elongation on 
the other, sometimes in a straight line, and sometimes in an 
elliptical curve, according to the different points of view in 

s Hind. Rev. Mr. Stoddard, a graduate of Yale College, missionary to the 
Nestorians, has repeatedly seen one of these bodies with the naked eye, from 
Mount Seir, near Oroomiah. Mr. Stoddard is known to the author as an excel- 
lent observer, and his testimony on this point may be fully relied on. 

f Mythological names were long since proposed for the satellites of Jupiter, 
viz., Io, Europa, Ganymede, Calisto ; but the mode of designating them by num- 
bers generally prevails. 



JUPITER. 



199 



which, we observe them from the earth. Their motion is al- 
ternately direct and retrograde ; they are sometimes stationary ; 
and, in short, they exhibit in miniature all the phenomena of 
the planetary system. Yarious particulars of the system are 
exhibited in the following table, the diameters being in miles, 
and the distances being taken from the center of the primary.* 



Satellite. 


Diameter. 


Distances. 


Sidereal Revolution. 


1 
2 
3 
4 


2,440 i 
2,190 
3,580 
3,060 


278,500 

443,300 

707,000 

1,243,500 


Id. 18h. 28m. 
3 13 15 
7 3 43 
16 16 32 



From this table we see that Jupiter's satellites are all larger 
than the moon, the second exceeding it by only 30 miles in di- 
ameter, the third by 1420 miles. The third, the largest of the 
whole, has still only ^th the diameter of the primary. The 
greater distances also of these moons compared with ours, 
reduces their apparent size and light as seen from Jupiter. 
Thus the largest of them would exhibit to a spectator on the 
equator of the planet, a diameter of only 36', which is only a 
little greater than that of the moon, while the smallest would 
appear only one-fourth as large. It is noticeable, that the 
satellites of Jupiter make very quick revolutions, when com- 
pared with the earth's moon, although they are all at greater 
distances from the primary than the moon from the earth ; the 
furthest revolving in about § of the moon's period, and the 
nearest more than 16 times as quick. This is because they are 
so powerfully attracted by the planet. To prevent their being 
drawn in from their circular orbits, a great projectile velocity is 
necessary. 

333. The orbits of the satellites are nearly or quite circu- 
lar, and deviate but little from the plane of the planet's 
equator, and of course are but slightly inclined to the plane of 
his orbit. They are, therefore, in a similar situation with 
respect to Jupiter as the moon would be with respect to the 
earth, if her orbit nearly coincided with the ecliptic, in which 



* Hind. 



200 THE PLANETS. 

case she would eclipse the sun every new moon, and be herself 
eclipsed every full moon. 

334. The eclipses of Jupiter's satellites, in their general 
conception, are perfectly analogous to those of the moon, but 
in their details they differ in several particulars. Owing to 
the much greater distance of Jupiter from the sun, and its 
greater magnitude, the cone of its shadow is more than sixty 
times that of the earth, stretching off into space more than 
55,000,000 miles. On this account, as well as on account of 
the little inclination of their orbits to that of their primary, the 
three inner satellites of Jupiter pass through the shadow and 
are totally eclipsed at every revolution. The fourth satellite, 
owing to the greater inclination of its orbit, sometimes, though 
rarely, escapes eclipse, and sometimes merely grazes the limits 
of the shadow, or suffers a partial eclipse.* These eclipses, 
moreover, are not seen by us, as is the case with those of the 
moon, from the center of their motion, but from a remote 
station, and one whose situation, with respect to the line of the 
shadow, is variable. This of course makes no difference in the 
times of the eclipses, but a very great difference in their visi- 
bility, and in their apparent situations with respect to the 
planet at the moment of their entering or quitting the shadow. 

335. The eclipses of Jupiter's satellites present some curious 
phenomena, which will be best understood from a diagram. 
Let A, B, C (Fig. 65) represent the earth in different parts of 
its orbit, revolving from A, through D, to C and B. If a line, 
joining S and A, be produced to meet the concave sphere of 
of the heavens xy, it marks the place of opposition, while the 
earth is at A. Hence, Jupiter, in the figure, is represented 
east of the opposition. "When the earth arrives at D, Jupiter 
is in opposition; and when at C, he is west of opposition. 
Remembering now, that the satellites revolve .in the same 
direction as the earth, it is obvious that when Jupiter is east of 
opposition, the immersions are seen, but the emersions generally 
take place behind the planet and are not seen ; so that the 
eclipse, in this position of the bodies, always precedes the 

* Herschel's Ast., p. 285. 



JUPITER. 201 

occultation. But if the earth passes beyond D, to C, then 
Jupiter is west of opposition ; for the place of opposition would 
be in SC produced. And now, each satellite passes behind the 
planet, and thus suffers occultation before it reaches the 

Fig. 65. 




shadow, and is eclipsed; and, in entering the shadow, the 
satellites are generally behind the planet, while in leaving it, 
they are in sight ; so that occupations precede eclipses. The 
fourth satellite often enters and leaves the shadow on the same 
side of the planet, and sometimes the third, and possibly the 
second, but this is never true of the first. 

336. "When one of the satellites is passing between Jupiter 
and the sun, it casts its shadow upon its primary, as the moon 
does on the earth in a solar eclipse, which is seen, by the tele- 
scope, traveling across the disk of Jupiter, as the shadow of the 
moon would be seen to traverse the earth by a spectator favor- 
ably situated in space. When Jupiter is east of opposition, 
the earth being at A (Fig. 65), the shadow strikes the disk of 
the planet, before the satellite itself is seen upon it, because 
the satellite will evidently come between S and Jupiter 
sooner than it does between A and Jupiter. Just the re- 
verse of this takes place, when the earth is at C, that is, 
when Jupiter is west of opposition. A satellite will then come 
between C and the planet, sooner than between S and the 
planet. We do not usually see the satellite itself projected on 



202 THE PLANETS. 

the disk of the primary, for, being illuminated like the prima- 
ry, it is not readily distinguishable from it ; but sometimes, 
when it happens to be projected on one of the belts, it is seen 
as a brig/it spot, making its transit across the disk. Occasion- 
ally, also, it is seen as a dark spot of smaller dimensions than 
the shadow. This curious fact has led to the conclusion that 
certain of the satellites have sometimes on their own bodies, or 
in their atmospheres, obscure spots of great extent.* 

337. A very singular relation subsists between the mean 
motions of the three first satellites of Jupiter. The mean longi- 
tude of the first, plus twice that of the third, minus three times 
that of the second, always equals 180 degrees. A curious 
consequence of this relation is, that the three satellites can 
never be all eclipsed at the same time; for then, having sev- 
erally the same longitude as the primary, their longitudes 
would be equal, and that of the first, plus twice that of the 
third, minus three times that of the second, would be nothing, 
and of course could not be 180 degrees. f The longitudes here 
mentioned are such as would be observed by a spectator on 
Jupiter, and not a spectator on the earth. 

338. The discovery of the system of Jupiter and his satel- 
lites, soon after the invention of the telescope, lent a powerful 
support to the Copernican system of astronomy, then just be- 
ginning to be received by astronomers, since it presented to 
the eye an exact miniature of the solar system, and exhibited 
an actual model of that arrangement of the sun and planets 
which had before only been contemplated by the eye of the 
mind ; and the laws of the planetary system, discovered by 
Kepler, were here actually seen to be verified, in the motions 
of this miniature system. Moreover, the eclipses of Jupiter's 
satellites, furnished one of those instantaneous events, occur- 
ring at the same moment of absolute time wherever seen, 
which are available for finding the longitudes of different 
places; and at that period it was deemed a more eligible 
method of determining this great practical problem of astrono- 
my than any method then in use. 

* Sir J. Herschel. t Biot. 



JUPITER. 203 

339. The eclipses of these satellites seem to have various 
requisites for determining longitudes, being, as already re- 
marked, seen at the same moment at all places where the 
planet is visible, being wholly independent of parallax, and 
being predicted beforehand with great accuracy the instant 
they occur at Greenwich, and given in the Nautical Almanac : 
but several circumstances conspire to render this method of 
finding the longitude less eligible than several other methods 
at present in use. The extinction of light in the satellite at its 
immersion, and the recovery of its light at its emersion, are 
not instantaneous, but gradual ; for the satellite, like the moon, 
occupies some time in entering into the shadow or in emerging 
from it, which occasions a progressive diminution or increase 
of light. The better the light afforded by the telescope with 
which the observation is made, the later the satellite will be 
seen at its immersion, and the sooner at its emersion.* In 
noting the eclipses even of the first satellite, the time must be 
considered as uncertain to the amount of 20 or 30 seconds; 
and those of the other satellites involve still greater uncer- 
tainty. Two observers, in the same room, observing with 
different telescopes the same eclipse, will frequently disagree 
in noting its time to the amount of 15 or 20 seconds, and the 
difference will always be the same way.f Better methods, 
therefore, of finding the longitude are now employed, although 
the facility with which the necessary observations can be made, 
and the little calculation required, still render this method eli- 
gible in many cases where extreme accuracy is not required. 
As a telescope is essential for observing an eclipse of one of 
these satellites, this method can not be practiced at sea. 

340. The grand discovery of the progressive motion of light 
was first made by observations on the eclipses of Jupiter's sat- 
ellites. Tn 1675, Roemer, a Danish astronomer, noticed that 
on comparing the average intervals and the true intervals be- 
tween the eclipses of a certain satellite, the true intervals are 
always shorter when the earth is approaching the planet, so 
that at the nearest point, the eclipse occurs 16m. 2 6 s . 6 too 



* This is the reason why observers are directed, in the Nautical Almanac, to 
use telescopes of a certain power . j Woodhouse, p. 840. 



204 THE PLANETS. 

soon. And, as the earth departs from Jupiter, Roemer per- 
ceived that the eclipses occur later than they should do accord- 
ing to the average interval, and that they are 16m. 26\6 too 
late, when at the greatest distance. He attributed this effect 
to the progressive motion of light. Dividing 190,000,000 
miles, the diameter of the earth's orbit, by 16m. 26 s .6, the 
time of crossing, he found 192,600 miles per second to be the 
velocity of light. This seemed, at first, quite incredible, and 
was received with distrust. But its correctness was soon es- 
tablished, by the discovery of the aberration of the stars, 
which gives the same result. 

341. Saturn comes next in the series as we recede from 
the sun, and has, like Jupiter, a system within itself, on a 
scale of great magnificence. In size it is, next to Jupiter, the 
largest of the planets, being 79,000 miles in diameter, or nearly 
10 times as large as the earth in diameter, and about 1000 
times as large in volume. It has likewise belts on its surface, 
and is attended by eight satellites. But a still more wonderful 
appendage is its Ring, a broad wheel encompassing the planet 
at a great distance from it. We have already intimated that 
Saturn's system is on a grand scale. As, however, Saturn is 
distant from us nearly 900,000,000 miles, we are unable to ob- 
tain the same clear and striking views of his phenomena that 
we do of the phenomena of Jupiter, although they really pre- 
sent a more wonderful mechanism. The disk of Saturn was 
described by Sir William Ilerschel as having the form of a 
rectangle with rounded corners ; but refined measurements, 
more recently made, show that it is an ellipse, and the planet, 
therefore, an ellipsoid. Its equatorial exceeds its polar diam- 
eter by about one-tenth.* 

The belts of Saturn, although clearly discerned by a good 
telescope, are far more indistinct than those of Jupiter. Spots, 
which occasionally appear on the belts, have enabled astrono- 
mers to determine the time of the diurnal rotation of Saturn, 
which is found to be about ten hours and a half (lOh. 29m.). 

342. When viewed with a good telescope, the body of the 

* Hind. 



SATURN. 205 

planet is seen to be surrounded by a broad thin ring, placed 
obliquely to our line of vision, so that it appears elliptical, 
showing one part in front of the planet and having a part hid 
behind it. At the ends of the ellipse, and sometimes entirely 
around it, is seen a division of the ring into two parts, of which 
the inner is the broadest. Though this division, on account of 
its immense distance, appears as a delicate line, yet it is, in 
reality, an interval of 1800 miles. The dimensions of the 
whole system, in round numbers, are as follows :* 

Miles. 

Diameter of the planet, 79,000 

From the surface of the planet to the inner ring, 20,000 

Breadth of the inner ring, 17,000 

Interval between the rings, 1,800 

Breadth of the outer ring, 10,500 

Extreme dimensions from outside to outside, . 176,000 

The figure (Frontispiece) represents Saturn, as it appears to 
a powerful telescope, surrounded by its rings, and having its 
body striped with dark belts, somewhat similar, but broader 
and less strongly marked than those of Jupiter, and owing, 
doubtless, to a similar cause, f The ring is opaque, since it 
casts a deep shadow on the planet. The earth is generally so 
situated that we can see some part, both of the shadow cast by 
the ring on the planet, and by the planet on the ring. 

The rings of Saturn have been long and carefully observed 
by Prof. G*. P. Bond, with the celebrated Cambridge refractor. 
He finds that there is a third ring within the others, which re- 
flects a dim light, and is about half as wide as the inner bright 
ring, but not separated from it by a noticeable interval. He 
also finds the two bright rings to be divided at times into sev- 
eral rings, with very narrow intervals. . In one instance the 
outer ring was divided into two, in another, into four parts ; 
and the inner one, in some cases, into six or eight delicate 
rings. The dark separating lines are usually traceable only at 
the ends of the ellipse. The investigations of Professor Peirce, 
aided by Bond's observations, have nearly, if not fully, estab- 
lished the fact of the fluidity of the rings, or at least, of the 

* Professor Struve, Mem. Art. Soc, iii., p. 301. f Sir J. Herscbel. 



206 THE PLANETS. 

incohereftcy of their parts, if they consist of solid matter. A 
changeableness of form and condition is indicated, also, when 
the ring presents to us its edge view. It sometimes appears 
as a delicate, uniform line ; at others, as a line of unequal 
thickness, whose thinner parts occasionally become entirely 
invisible ; and the inequalities change their aspects, from time 
to time, in a manner not to be accounted for by the revolution 
which the system is known to have. 

A most remarkable fact relating to the rings, is their exceed- 
ing thinness. They have generally been regarded as about 100 
miles thick. But Bond's observations lead to the conclusion 
that their thickness is less than 40 miles. If a model of the 
rings, one foot in diameter, were cut out of common writing- 
paper, the thickness would be too great to represent them 
properly. 

343. Saturn's ring, in its revolution around the sun, always 
remains parallel to itself. 

If we hold, opposite to the eye, a circular ring or disk, like 
a piece of coin, it will appear as a complete circle when it is 
at right angles to the axis of vision ; but when oblique to that 
axis, it will be projected into an ellipse more and more narrow, 
as its obliquity is increased, until, when its plane coincides 
with the axis of vision, it is projected into a straight line. Let 
us place on the table a lamp or a ball, to represent the sun, 
and, holding the ring at a certain distance, inclined a little 
toward the central body, let us carry it round, always keeping 
it parallel to itself. During its revolution it will twice present 
its edge to the lamp or ball at opposite points, and twice, at 
90° distance from those points, it will present its broadest face 
toward the central body. At intermediate points it will ex- 
hibit an ellipse more or less open, according as it is nearer one 
or the other of the preceding positions. It will be seen, also, 
that in one-half of the revolution the lamp shines on one side 
of the ring, and in the other half of the revolution on the other 
side. Such would be the successive appearances of Saturn's 
ring to a spectator on the sun ; and since the earth is, in re- 
spect to so distant a body as Saturn, very near the sun, those 
appearances are presented to us nearly in the same manner as 
though we viewed them from the sun. Accordingly, we some- 



SATURN. 



207 



times see Saturn's ring under the form of a broad ellipse, 
which grows continually more and more narrow until it passes 
into a line, and we either lose sight of it altogether, or, with 
the aid of the most powerful telescopes, we see it as a fine line 
drawn across the disk, and projecting out from it on each side. 
As the whole revolution occupies nearly 30 years, and the plane 
of the ring passes across the earth's orbit twice in the revolu- 
tion, the phenomena attending the edge view occur every 15 
years, when the ring may within a single year disappear re- 
peatedly, and for different reasons, as described in Art. 346. 

344. The learner may perhaps gain a clearer idea of the 
foregoing appearances from the following diagram. 

Let A, B, C, &c, Fig. 66, represent successive positions of 
Saturn and his ring in different parts of his orbit, while db 

Fig. 66. 




represents the orbit of the earth.* Were the ring when at C 
and G perpendicular to the line joining CG, it would be seen 
by a spectator situated at a or h as a perfect circle, but being 
inclined to the line of vision 28° ll 7 , it is projected into an 
ellipse. This ellipse gradually contracts to a straight line, as 
the ring passes to A and E, where its nodes are in the direc- 
tion of the earth's orbit. From E to G the ellipse widens 
again, till at G the breadth is nearly half as great as its length. 
Through the quadrant from G to A it contracts, and from A 
to C it expands, occupying about 7£ years in going from the 



• © It may be remarked by tbe learner, that these orbits are made so elliptical, 
not to represent the eccentricity of either the earth's or Saturn's orbit, but merely 
as the projection of circles seen very obliquely. 



208 



THE PLANETS. 



maximum breadth to the minimum, and the reverse. These 
changes are visible in a telescope of moderate powers, except 
that the ring is too thin to be seen at A and E. 

345. There are three ways in which the ring may fail to be 
visible during the period when the line of its nodes is crossing 
the earth's orbit. 1st. It may present its edge exactly to the 
earth, when in common telescopes, it subtends too small an 
angle to be seen. 2d. It may present its edge exactly to the 
sun, so that neither side of the ring is enlightened. 3d. Its 
plane may be directed between the earth and sun, when the 
dark side is toward us. The two first causes may be consid- 
ered as only momentary ; for the plane of the ring passes the 
breadth of the sun in less than two days, and of the earth in 
about 20 minutes. But the third cause may continue to ren- 
der the ring invisible for several months. 

In Fig. 67 let S be the sun, C the place of Saturn, when the 



Fig. 67. 




nodes of its ring are in the line CS, passing through the sun. 
Let EFGH be the earth's orbit; then EB, GD, parallel to SO, 
will include BD, the arc of Saturn's orbit in which the ring 
will present its edge to the earth's orbit. If the orbits are 
supposed to be in the plane of the paper, we must conceive the 
small ellipses at B, C, D, representing the ring, to be inclined 
about 28°, having their section with the orbit, in BE, CS, and 
DG, respectively. While Saturn passes over the arc BD, the 
plane of its ring will pass once through the sun, and may re- 
peatedly pass through the earth, on account of the revolution 
of the latter. The time of describing BD differs only about 
six days from a year ; for, since Saturn is 9.54 times as far 



SATURN. 209 

from the sun as the earth is, therefore, 9.54 : 1 : : SB : SE : : 
rad : sin SBE ; which is thus found to be 6° 1'. Hence, BSO 
= 6° 1', and BSD=12° 2'. But Saturn describes 12° 2' in 359£ 
days, near six days less than a year. The earth therefore very 
nearly describes the orbit EFGH, while Saturn describes the 
arc BCD. 

346. The disappearances of the ring may be variously 
modified during the year of passing the node, according to the 
place of the earth when the nodal line first reaches its orbit. 
A few cases are here described. If the earth is at G, when the 
plane of the ring passes E, then, while the nodal line moves 
from BE to CS, the earth will go from G through H to E, and 
must cross that line below H. The ring will begin its disap- 
pearance at that moment ; but the disappearance will continue, 
as the earth proceeds to E, because the dark side is toward it. 
This disappearance will last about two months, and close when 
the plane of the ring at C passes the sun ; for after that, the 
illuminated side will be toward the earth. While the earth 
proceeds from E through F to G, the plane of the ring will 
move from CS to DG, before the earth can overtake it. 

Again, if the earth had advanced some distance on the 
quadrant GH, when the ring's line of nodes was at E, then 
they will meet further from H than before, say at M ; after 
which the dark side will be toward the earth. The plane of 
the ring will pass the sun, when the earth is on the quadrant 
EF, after which the bright side is presented to the earth. But 
the earth will overtake the nodal line before reaching G, and 
therefore look again upon the dark side, until it recrosses the 
line on the quadrant GH. Thus there are two periods of dis- 
appearance. These two may possibly unite in one, of 8 
months' continuance ; this will happen when the earth and 
node-line pass F at the same time ; for then the plane of the 
ring is between the earth and sun both before and after pass- 
ing through the sun. 

It is a possible thing, that no disappearance at all should 
happen during the nodal year. Suppose the earth at F, when 
the line of nodes arrives at E. Then, while the line moves 
from BE to CS, the earth will describe FGH, all the time on 
the luminous side of the ring ; the earth and sun will both bn 

14 



210 THE PLANETS. 

in the line CS at once, the planet being in conjunction with 
the snn; and, after the earth has passed H toward F, the 
bright side of the ring is again in view. Thus, there is only a 
momentary disappearance, and that when the planet itself is 
lost in the blaze of the sun's light. 

But in general there are two periods of disappearance within 
the nodal year, arising from the third cause (Art. 345), each 
beginning and ending with a disappearance from the first or 
second cause. 

347. The rings of Saturn must present a magnificent spec- 
tacle to that hemisphere of the planet to which their illuminated 
side is turned, appearing as bright arches several degrees in 
width, and spanning the sky from one side of the horizon to 
the other. They revolve diurnally in the same plane as the 
planet itself, and in about the same time, 10^ hours. 

348. Saturn is attended by eight satellites. Their sizes 
vary from 500 to 2,850 miles ;* but, on account of their great 
distance, they are seen only with the best instruments. They 
are all external to the rings, and the eighth at the distance of 
2,500,000 miles from the planet.f The seventh was discovered 
in Cambridge, Mass., by Professor Bond. Their orbits are 
nearly in the plane of the ring, and make an angle of about 
28° with the orbit of the planet. Only the two interior ones 
are eclipsed, except when the ring is seen edgewise.;): 

349. Uranus, the next planet in the series, was discovered 
by Sir William Herschel, in 1781. Previous to this time, 
Saturn had, from a high antiquity, been considered li the 
outermost boundary of the solar system ; but this discovery 
doubled the dimensions of the system, bringing to light a large 
planet at about twice the distance of Saturn from the sun, and 
about 19 times the distance of the earth, or 1800 millions of 
miles. It was named by the discoverer the Georgian^ in honor 
of his patron, George III. ; but this name being unacceptable 

* Hind. 

f They have received the names, Mimas, Enceladus, Tethys, Dione, Rhea, 
Titan, Hyperion, and Japetus. 
X Sir J, HerscheL 



URANUS NEPTUNE. 211 

to astronomers of other countries, the planet was called Herschel 
in America, after the name of the discoverer, and Uranus* on 
the continent of Europe, which last appellation is now uni- 
versally adopted. The diameter of Uranus is about 35,000 
miles, and consequently its volume more than 80 times that of 
the earth. Its revolution around the sun occupies nearly 84 
years, so that its position among the stars varies but little for 
several years in succession, since it shifts its place only a little 
more than four degrees in a year, and of course would remain 
in the same sign of the Zodiac seven years. Its path lies very 
near the ecliptic, being inclined to it less than 0° 47'. The sun 
himself, when seen from Uranus, dwindles almost to a star, 
subtending, as it does, an angle of only V 40" ; so that the 
surface of the sun would appear there nearly 400 times less 
than it does to us. 

350. The satellites of Uranus are exceedingly minute ob- 
jects, and visible only to the most powerful telescopes. Al- 
though Sir "William Herschel assigned six satellites to this 
planet, yet only two of the number (the second and fourth in 
the order of distances) have, until quite recently, been seen by 
other astronomers. Two others have of late been added, and 
an increasing confidence is beginning to be felt that the entire 
number announced by Herschel will be identified. The orbits 
of these satellites, says Sir John Herschel, offer remarkable, 
and indeed quite unexpected and unexampled peculiarities. 
Contrary to the unbroken analogy of the whole planetary 
system, whether of primaries or secondaries, the planes of their 
orbits are nearly perpendicular to the ecliptic, being inclined 
no less than 78° 58' to that plane, and in these orbits their 
motions are retrograde. Instead of advancing from west to 
east, as is the case with every other planet and satellite, they 
move in the opposite direction, or from east to west. With 
this exception, all the motions of the planets, whether around 
their own axes or around the sun, and that of the sun himself 
on his axis, are from west to east. 

351. Neptune is (so far as is known) the last planet of the 

* From ovpavot, the father of Saturn. 



212 THE PLAKETS. 

series, being removed from the sun to the immense distance of 
nearly 3000 millions of miles (2,862,457,000). Its diameter is a 
little less than that of Uranus, being 31,000 miles,* Its volume 
is nearly sixty times that of the earth. Its periodic time is 
164-! years, which is about twice that of Uranus. Its orbit is 
nearly circular, and but little inclined to the ecliptic (1° 47'). 

The discovery of the planet Neptune is the most remarkable 
astronomical event of our times, and is generally considered 
as the most extraordinary discovery ever made in physical 
science. The leading steps of the process were as follows. 
The planet Uranus had long been known to be subject to cer- 
tain irregularities in its revolution around the sun, not accounted 
for by all the known causes of perturbation. In some cases 
the deviation from the true place, as given by the tables, dif- 
fers from actual observation two minutes of a degree — a quan- 
tity indeed which seems small, but which is still far greater 
than occurs in the case of the other planets, and far too great 
to satisfy the extreme accuracy required by modern astronomy. 
This fact long since suggested to astronomers the possibility of 
one or more additional planets, hitherto undiscovered, which, 
by their attractions, exert on Uranus a great disturbing influ- 
ence. Le Yerrier, a distinguished French astronomer, assum- 
ing the existence of such a planet, applied himself, by the aid 
of the calculus, guided by the law of universal gravitation, to 
the inquiry where the hidden planet was situated — at what 
distance from the sun — and at what point of the starry heav- 
ens ? From Bode's law of the planetary distances (Art. 299), 
according to which Saturn is nearly twice as far from the sun 
as Jupiter, and Uranus twice as far as Saturn, he inferred that, 
if a planet exists beyond Uranus, its distance is probably about 
twice that of Uranus, or about 3,600 millions of miles from the- 
sun, which is nearly thirty-eight times that of the earth. He 
assumed it, however, to be thirty-six times the earth's mean 
distance. The corresponding periodic time would be 216 years. 
After reasoning from analogy, and the doctrine of universal 
gravitation, respecting the position and mass which a body 
must have in order to account for the perturbations of Uranus, 
equations were formed between these perturbations and the 



Hind. 



NEPTUNE. 213 

assumed and unassumed elements of trie body in question. 
These equations were exceedingly complex and difficult of re- 
duction ; but, by the most ingenious artifices, the several un- 
known quantities were successively eliminated, either directly 
or by repeated approximations, until the great geometer ar- 
rived at expressions for the elements of the unknown planet, 
which indicated its place among the stars, its quantity of mat- 
ter, the shape of its orbit, and the period of its revolution. 
Having placed the body in various positions in the orbit thus 
determined, he found that when situated at a point in the con- 
stellation Oapricornus, its effect upon Uranus would be such 
as corresponded to the irregularities to be accounted for ; that 
on the 1st of January, 1847, the hidden planet would have a 
longitude of 326° 32', and would lie about five degrees east- 
ward of the well-known star Delta Capricomi. He further 
asserted that it would have an apparent diameter <of about 3", 
and therefore be visible to large telescopes. 

352. Having communicated these results to tbe French 
Academy, at their sitting on the 31st of August, 1846, Le 
Yerrier soon afterward made them known to Dr. Galle, one of 
the astronomers of the Hoyal Observatory of Berlin, with the 
request that he would search for the stranger with the power- 
ful telescope at his command. On the same evening that Dr. 
Galle received the communication, namely, on the 23d of Sep- 
tember, he directed his telescope toward the spot assigned for 
the planef, and there it was, within less than a degree of the 
place indicated by Le Terrier, and having an apparent mag- 
nitude within half a second of that assigned. To show the 
near correspondence between theory and observation, we may 
remark that the predicted longitude, for the 23d of September, 
at midnight, was 324° 58', and the observed longitude was 
325° 5_2'.8 ; the predicted diurnal motion in longitude was 69", 
and the observed 74". These results struck the scientific world 
with astonishment, and their confirmation w r as one of the 
greatest achievements of the human mind. 

353. It has often happened, in the history of great discov- 
eries, that the same hidden truth is revealed simultaneously to 
different inquirers, and accordingly, by a singular coincidence. 



214 THE PLANETS. 

a young mathematician of the University of Cambridge (Eng.), 
Mr. Adams, had, without the least knowledge of what JVL Le 
Yerrier was doing, arrived at the same great result. But 
having failed to publish his paper until the world was made 
acquainted with the facts through the other medium, he has 
lost much of the honor which the priority of discovery would 
have gained for him. Thus two distinguished mathematicians, 
unknown to each other, and by entirely independent processes, 
had arrived at the same results, as regarded both the existence 
of the supposed planet, and the region of the starry heavens 
where at that moment it lay concealed; and, to crown all, 
astronomers, in obedience to the direction of one of them, had 
pointed their telescopes to the spot and found it there. The 
conviction on the mind of every one was, that nothing but 
absolute truth could abide a test so unequivocal. It still 
remained, however, to determine by observation whether the 
body actually conformed, in all respects, to the results of 
theory. To settle this point completely, that is, to determine 
with precision the elements of the orbit from observation, 
would require a long time in a planetary body whose motion 
was so slow that more than two centuries, as was supposed, 
would be required to complete a single revolution. But if it 
should be found that, among preceding catalogues of the stars, 
this body might have been included, and its place recorded as 
a fixed star, then, by comparing that place with its present 
position, and noting the interval of time between the two 
observations, we might thus learn the rate of its motion, and 
its periodic time, and might thence deduce various other par- 
ticulars dependent on these elements. Our distinguished coun- 
tryman, Mr. Sears C. Walker, then connected with the observ- 
atory at Washington, undertook this investigation. First, 
from the observations already accumulated, he calculated the 
path which the planet must have pursued for the last fifty or 
sixty years, and by tracing this path among the stars of La- 
lande's catalogue, he found that it passed within two minutes 
of a star of the seventh magnitude, which was recorded as 
being seen in May, 1795. Professor Hubbard, of the same 
observatory, on reconnoitering for this star, found that it was 
missing. Little doubt remained that the star seen by Lalande 
was the planet of Le Yerrier; and this conclusion was con 



MOTIONS OF THE PLANETARY SYSTEM. 215 

firmed by calculating its orbit on this supposition, and com- 
paring the results with the places it has actually occupied since 
it fell within the sphere of observation. The results thus 
obtained, however, were materially different from those of Le 
Terrier and Adams. Instead of a period of 216 years, they 
give only a period of 164^ years ; and instead of a distance of 
3,600 millions of miles, the new period would require a distance 
of only 2,862 millions. The eccentricity of the orbit, moreover, 
according to Walker, is much less than had been assigned to 
it, the orbit being in fact very nearly circular, while, by Le 
Terrier's estimate, it was considerably elliptical. The longi- 
tude, in fact, proved to be nearly the same as that assigned to 
it. But this close agreement is to be considered accidental ; 
for Le Terrier himself fixed on the precise place which he 
named, as only the most probable. On account of uncertainty 
in the data, he stated that there might be a variation of 9° 
either way.* The elements thus corrected account fully and 
completely for the irregularities of Uranus sought to be ex- 
plained, within a single second, as determined by Professor 
Peirce.f 

MOTIONS OF THE PLANETARY SYSTEM. 

354. We have waited until the learner maybe supposed 
to be familiar with the heavenly bodies individually, before 
inviting his attention to a systematic view of the planets in 
their revolutions around the sun, and their grand laws. 

There are two methods of arriving at a knowledge of the 
motions of the. heavenly bodies. One is, to begin with the 
apparent, and from these to deduce the real motions; the 
other is to begin with considering things as they really are in 
nature, and then to inquire why they appear as they do. The 
latter of these methods is by far the more eligible. It is much 
easier than the other ; and proceeding from the less difficult to 
that which is more so — from motions which are very simple to 
such as are complicated, it finally puts the learner in possession 
of the whole machinery of the heavens. We shall in the first 
place, therefore, endeavor to introduce the student to an ac- 

* Loomis's Recent Prog, of Astron., pp. 50-52. 

f Amer. Journal of Science, New Series, vol. v., p. 436. 



216 THE PLANETS. 

quaintance with the simplest motions of the planetary system, 
and afterward to conduct him gradually through such as are 
more complicated and difficult. 

355. When viewed from the center of their motions, the 
revolutions of the planets would appear simple and harmonious, 
all coursing round the spectator from west to east in regular 
order, in nearly the same great highway, though with very 
different degrees of velocity. Let us, then, suppose ourselves 
standing on the sun, and contemplate the revolutions of the 
planets, first, severally, and then as forming one grand whole, 
consisting of numerous parts, but bound together under the 
same laws in one vast empire. We should see Mercury making 
very perceptible progress around the heavens, like the moon in 
its motions about the earth, his rate of motion eastward being 
about one-third as rapid as that of the moon, since he com- 
pletes his entire revolution in about three months. It will, at 
first, aid our conceptions of the respective positions of the 
planetary orbits, to imagine the ecliptic to be marked out on 
the face of the visible heavens in a palpable line distinctly vis- 
ible to the eye. If we, stationed at the sun, watch the motions 
of Mercury, we shall see it cross the ecliptic in two opposite 
points of the heavens, constituting its nodes; and we shall see 
it, when half way between the nodes, at an angular distance 
from the ecliptic of about 7°, this being the inclination of its 
orbit. Knowing the position of the orbit of Mercury with 
respect to the ecliptic, we may now, in imagination, represent 
that orbit in a great circle passing through the center of the 
planet and the center of the sun, and cutting the plane of 
the ecliptic in two opposite points in an angle of 7 de- 
grees. The paths of both planets appear as circles among 
the stars; and if we suppose them to be visibly traced on 
their respective planes, the observer, while at the sun, can 
not distinguish them from circles, of which he is the cen- 
ter. But, if we imagine him transported to a great distance 
from the sun, in a line at right angles to the ecliptic, he will 
discern the true forms of the orbits. The earth's orbit can not 
even now be distinguished from a circle, but the sun is plainly 
a little out of its center. The orbit of Mercury, however, is 
distinctly elliptical, with the sun in one of its foci. On his 



MOTIONS OF THE PLANETARY SYSTEM. 217 

return to liis station at the sun, he loses all perception of this 
elliptical form, and of his eccentric position. But he perceives 
the consequences of these facts, — an alternate increase and de- 
crease of size and of velocity in the planets. The earth slowly 
becomes larger, and moves more swiftly, till it reaches a cer- 
tain position, and then diminishes both in size and velocity, 
and attains a minimum at a point 180° from the maximum. 
Mercury passes through still greater changes of the same kind. 

356. A clear understanding of the motions of Mercury, and 
of the relations of its orbit to the plane of the ecliptic, will ren- 
der it easy to understand the same particulars in regard to each 
of the other planets. Standing on the sun, we should see each 
of the planets pursuing a similar course to that of Mercury, all 
moving from west to east, differing from each other chiefly in 
two respects, namely, in their velocities, and in the distances 
to which they recede from the ecliptic, or their inclinations. 
We have supposed the observer to select the plane of the earth's 
orbit as his standard of reference, and to see how each of the 
other orbits is related to it ; but such a selection of the ecliptic 
is entirely arbitrary ; the spectator on the sun, who views the 
motions of the planets as they actually exist in nature, would 
make no distinction between the different orbits, but merely 
inquire how they are mutually related to each other. Taking, 
however, the ecliptic as the plane to which all the others are 
referred, we do not, as in the case of the other planets, inquire 
how its plane is inclined, nor what are its nodes, since it has 
neither inclination nor node. 

357. Such, in general, are the real motions of the planets, 
and such the appearances which the planetary system would 
exhibit to a spectator at the center of motion. But, in order 
to represent correctly the positions of the planetary orbits, at 
any given time, three things must be regarded : the Inclination 
of the orbit to the ecliptic ; the position of the line of the 
Nodes, and the position of the line of the Apsides. In our 
common diagrams, the orbits are incorrectly represented, be- 
ing all in the same plane, as in the following diagram, where 
AEB (Fig. 68) represents the orbit of Mercury as lying in the 
same plane with the ecliptic. To exhibit its position justly, 



218 



THE PLANETS. 



AB being taken as the line of the nodes, the plane should be 
elevated on one side about 7°, and depressed the same number 
of degrees on the other side, turning on the line AB as on a 
hinge. But even then the representation may be incorrect in 
other respects, for we have taken it for granted that the line 
of the nodes coincides with the line of the apsides, or that the 
orbit of Mercury cuts the ecliptic in the line AB, the major 
axis of the orbit, whereas it may lie in any given position with 
respect to the line of apsides, according to the longitude of the 
nodes. If, for example, the line of nodes had chanced to pass 

Fig. 68. 




through Taurus and Scorpio instead of Cancer and Capricorn, 
then it would have been represented by the line b fll instead of 
the line passing through S>, and the plane, when elevated or 
depressed with respect to the plane of the ecliptic, would be 
turned on this line in our figure. Moreover, our diagram rep- 
resents the line of apsides as passing through Cancer and Cap- 
ricorn, whereas it may have any other position among the 
signs, according to the longitudes of the perigee and apogee. 



358. Having acquired as correct an idea as we are able of 
the planetary system, as seen from the sun, and of the posi- 
tions of the orbits with respect to the ecliptic, let us next in- 



MOTIONS OF THE PLANETARY SYSTEM. 219 

quire into the nature and causes of the apparent motions. The 
apparent motions of the planets are exceedingly unlike the real 
motions, a fact which is owing to two causes : first, we view 
them out of the center of their orbits ; secondly, we are our- 
selves in motion. From the first cause, the apparent places of 
the planets are greatly changed by perspective ; and, from the 
second cause, we attribute to the planets changes of place 
which arise from our own motions, of which we are uncon- 
scious. 

359. The situation of a heavenly body, as seen from the 
center of the sun, is called its heliocentric place ; as seen from 
the center of the earth, its geocentric place. The geocentric 
motions of the planets must, according to what has just been 
said, be far more irregular and complicated than the heliocen- 
tric, as will be evident from the following diagram, which 
represents the geocentric motions of Mercury for two entire 
revolutions, embracing a period of nearly six months. Let S 
(Fig. 69) represent the sun, 1, 2, 3, &c, the orbit of Mercury, 
a, b, c, &c, that of the earth, and GT the concave sphere of the 
heavens. The orbit of Mercury is divided into 12 equal parts, 
each of which he describes in 7J days ; and a portion of the 
earth's orbit described by that body in the time that Mercury 
describes the two complete revolutions, is divided into 21 
equal parts. Let us now suppose that Mercury is at the point 
1 in his orbit, when the earth is at the point a ; Mercury will 
then appear* in the heavens at A. In 7 J days Mercury will 
have reached 2, while the earth has reached h, when Mercury 
will appear at B. By laying a ruler on the point c and 3, d 
and 4, and so on, in the order of the alphabet, the successive 
apparent places of Mercury in the heavens will be obtained. 
From A to C, the apparent motion is direct, or in the order of 
the signs ; from C to G it is retrograde ; at G it is stationary a 
while, and then direct through the whole arc GT. At T the 
planet is again stationary, and afterward retrograde along the 
arc TX. Hence it appears that the motions of an inferior 
planet, as viewed from the earth, are exceedingly irregular and 
complicated, although it is all the while pursuing its course at 
a nearly uniform rate, and in the same unvarying direction 
around the sun. It moves forward when near the superior 



220 



THE PLANETS. 



conjunction, backward when near the inferior, and is station- 
ary near the points of greatest elongation. The planet moves 
sometimes very slowly, and then rapidly ; at one time back- 
ward over a small space, and then forward for a great distance. 





Yet all these apparent irregularities are owing to the two 
causes already adverted to, viz., the effects produced by per- 
spective, and by the motions of the spectator himself. Yenus 
exhibits a variety of motions similar to those of Mercury, ex- 
cept that the changes do not succeed each other so rapidly, 
since her period of revolution approaches more nearly to that 
of the earth. 



360. The apparent motions of the superior planets are, like 



MOTIONS OF THE PLANETARY SYSTEM. 



221 



those of Mercury and Yenns, alternately direct and retrograde, 
and between the two the planets are stationary. In this case, 
however, the earth moves faster than the planet, and the 
planet has its opposition, but no inferior conjunction ; whereas 
an inferior planet has its inferior conjunction, but no opposi- 
tion. These differences render the apparent motions of the 
superior planets in some respects unlike those of Mercury and 
Yenus. On the side of the sun most remote from the earth, 
the motion of a superior planet is direct, because, as is the case, 
with Yenus in her superior conjunction (see figure 61), the 
only effect of the earth's motion is to accelerate it ; but when 
the planet is in opposition, the earth is moving past it with 
greater velocity, and makes the planet seem to move back- 
ward, like the apparent backward motion of a vessel when we 
overtake it and pass rapidly by it in a steamboat. 

361. Let ABCD (Fig. 70) represent the earth in different 
positions in its orbit, M a superior planet as Mars, and NR an 




arc of the concave sphere of the heavens. First, suppose the 
planet to remain at rest in M, and let us see what apparent 
motions it would receive from the real motions of the earth. 
When the earth is at B, it will see the planet in the heavens 



222 THE PLANETS. 

at N" ; and as the earth moves successively through CDEF, the 
planet will appear to move through OPQK ; B and F are the 
two points of greatest elongation of the earth from the sun, as 
seen from the planet : between these two points, while passing 
through the part of its orbit most remote from the planet (at 
which time the planet is seen in superior conjunction), the 
earth, by its own motion, gives an apparent motion to the 
planet in the order of the signs ; that is, the apparent motion 
given by the earth's motion, when the planet is seen toward 
its superior conjunction, is direct. But in passing from F to B 
through A, when the planet is seen toward its opposition, the 
apparent motion given to the planet by the earth's motion is 
retrograde. But the superior planets are not in fact at rest, as 
we have supposed, but are all the while moving eastward, 
though with a slower motion than the earth. Indeed, with 
respect to the remotest planets, as Saturn and Uranus, the for- 
ward motion is so exceedingly slow, that each remains for a 
long time in the same sign of the zodiac. Still, the effect of 
the real motions of all the superior planets eastward, is to in- 
crease the direct apparent motion communicated by the earth, 
and to diminish the retrograde motion, as will be readily seen 
from the figure. 



tlHAPTEK XI. 

DETERMINATION OF THE PLANETARY ORBITS 

ERIES ELEMENTS OF THE ORBIT OF A PLANET QUANTITY OF 

MATTER IN THE SUN AND PLANETS. 

362. In Chapter IT. we have shown that the figure of the 
earth? s orbit is an ellipse, having the sun in one of the foci, 
and that the earth's radius describes equal spaces in equal 
times ; and in Chapter III. we have remarked that these are 
only particular examples under the law of Universal Gravita- 
tion, as is also the additional fact, that the squares of the 
periodic times of the planets are as the cubes of the major 



DETERMINATION OF THE PLANETARY ORBITS. 223 

axes of their orbits. We may now learn more particularly 
the process by which the illustrious Kepler was conducted to 
the discovery of these grand laws of the planetary system. 
From the apparent motions of the heavenly bodies, as seen 
projected on the face of the sky, the ancient astronomers in- 
ferred that their orbits were necessarily circular, and the mo- 
tions actually uniform. Still, Hipparchus and Ptolemy were 
not ignorant of the fact that the sun moves faster through the 
winter than through the summer signs, performing the half 
of his revolution around the earth nearly eight days sooner 
from the autumnal to the vernal, than from the vernal to the 
autumnal equinox. This led them to infer that the earth is 
not in the center of the circle, but nearer to one side of the 
circle than to the other, by which means the sun would appear 
to move more rapidly in that part of its orbit than in the op- 
posite part, just as a steamboat appears, to a spectator on the 
shore, to move faster when nearer than when more remote 
from the shore, although her actual speed is the same in both 
cases. On a similar supposition Tycho Brahe made a great 
number of very accurate observations on the planetary motions, 
which served Kepler as standards of comparison for results, 
which he deduced from calculations, founded on the applica- 
tion of geometrical reasoning to various hypotheses which he 
successively assumed as to the figure of the planetary orbits; 
first supposing the orbit to be of a certain figure, then determ- 
ining from the geometrical properties of the curve what mo- 
tions the body would appear to us to have when moving in 
such a path, and finally testing his conclusions by comparing 
them with the facts, as determined by Tycho, from observa- 
tion. 

363. Kepler first applied himself to investigate the figure 
of the orbit of Mars, the motions of which planet appeared 
more irregular than those of any other planet except Mercury, 
which, being seldom seen, had been very little studied. Like 
Ptolemy and Tycho, he first supposed the orbit to be circular, 
and the planet to move uniformly about a point at a certain 
distance from the sun. He made seventy suppositions before 
he obtained one that agreed with observation, the calculation 
of which was extremely long and tedious, occupying him more 



224: THE PLANETS. 

than five years.* The supposition of an equable motion in a 
circle, however varied, could not be made to conform to the 
observations of Tycho, whereas the supposition that the orbit 
was an oval figure, depressed at the sides, but coinciding with 
a circle at the perihelion, agreed so nearly with observation as 
to leave no doubt that the orbit of Mars is an ellipse, having 
the sun in one of its foci. He immediately inferred that the 
same is true of the orbits of all the other planets ; and a simi- 
lar comparison of this hypothesis with observation, confirmed 
its truth. Thus he established the first great law, viz., The 
planets revolve about the sun in ellipses, having the sun in one 
of the foci. 

364. Kepler also discovered, from observation, that the 
velocities of a planet, when in the apsides of its orbit, are in- 
versely as the distances, and therefore the product of the ve- 
locity into the distance would, in those two points, make the 
same quantity. But the velocity is the length of arc described 
in a unit of time ; and the length of an arc multiplied by its 
radius, is double the sector upon that arc. Therefore the area 
described by the radius vector at one apsis equals that de- 
scribed at the other apsis in the same time. (Fig. 32, p. 86.) 
Although he could not prove, from observation, that the same 
was true in every point of the orbit, yet analogy suggested 
that such was probably the fact. Therefore, assuming this 
principle as true, and hence deducing the equation of the 
center (Art. 200), he found the result to agree with observa- 
tion, and thus arrived at the conclusion (which has since been 
proved true (Art. 181) from the principles of common mechan- 
ics), that the radius vectors of the planetary orbits describe about 
the sun equal areas in equal times. 

365. Having in his researches, that led to the discovery of 
the first of the above laws, found the relative mean distances 
of the planets from the sun (Art. 308), and knowing their 

* Logarithms were invented during the age of Kepler, but were not available 
to him until his most laborious calculations had been performed. In relation to 
these, he expresses himself thus : Si te hvjus laboriosce methodi perteesum fueril, jure 
mei te misereat, qui earn ad minimum septuagies ivi cum plurima temporis jactura ; et mirari 
desines hunt quintum jam annum abire, ex quo Martem aggressus sum. 



ELEMENTS OF THE PLANETARY ORBITS. 225 

periodic times from observation, Kepler next endeavored to 
ascertain if there was any relation between the distances and 
times of revolution, having a strong passion for tracing analo- 
gies in nature. He saw at once that the more distant a planet 
is from the sun, the slower it moves ; so that the periodic times 
of the remoter planets are increased on two accounts : first, 
because they have a longer path to traverse ; and secondly, 
because they actually move more slowly in their orbits than 
the planets nearer the sun. Saturn, for example, is 9-J times 
further from the sun than the earth is ; and since the circum- 
ferences of circles are as their radii, the orbit of Saturn must 
be larger than the earth's in the same ratio ; so that if the 
periodic time of Saturn were longer than the earth's, merely 
because its orbit is larger, that period would be 9 J years, 
whereas it is 30 years. Hence it is evident that the periodic 
times of the planets increase in a greater ratio than their dis- 
tances from the sun, but in a less ratio than the squares of the 
distances, for then the time of Saturn would be about 90 years. 
Kepler then compared the squares of the times with the cubes 
of the distances, and found an exact agreement between them. 
Thus he discovered the famous law, the squares of the periodic 
times of all the planets are as the cubes of their mean distances 
from the sun.* 

366. This law is strictly true only in relation to planets 
whose quantity of matter in comparison with that of the cen- 
tral body is inappreciable. When this is not the case, the 
periodic time is. shortened in the ratio of the square root of the 
sun's mass divided by the sun's plus the planet's mass, as ex- 
pressed by the formula (tjtt — / • ^he mass of the planets 

is, however, so small compared to the sun's, that this modifica- 
tion of the law is unnecessary except where extreme accuracy 
is required. 

ELEMENTS OF THE PLANETARY ORBITS. 

367. The particulars necessary to be known in order to 
determine the precise situation of a planet at any instant, are 

* Vince's Complete System, i., p. 98. 
15 



226 THE PLANETS. 

called the Elements of its Orbit. They are seven in number: 
of which the first two determine the position of the plane of 
the orbit, and the other five relate to the orbit and the planet 
in that plane. These elements are : 

1. The position of the line of the nodes. 

2. The inclination to the ecliptic. 

3. The periodic time. 

4. The m,ean distance from the sun, or semi-axis major. 

5. The eccentricity. 

6. The place of the perihelion. 

7. The place of the planet in its orbit at a particular epoch. 

368. It may at first view be supposed that we can proceed 
to find the elements of the orbit of a planet in the same man- 
ner as we did those of the solar or lunar orbit, namely, by ob- 
servations on the right ascension and declination of "the body, 
converted into latitudes and longitudes by means of spherical 
trigonometry (see Art. 132). But in the case of the moon, 
we are situated in the center of her motions, and the apparent 
coincide with the real motions : and in respect to the sun, our 
observations on his apparent motions give us the earth's real 
motions, allowing 180° difference in longitude. But as we 
have already seen, the motions of the planets appear exceed- 
ingly different to us, from what they would if seen from the 
center of their motions. It is necessary, therefore, to deduce 
from observations made on the earth the corresponding results 
as they would be if viewed from the center of the sun; that is, 
in the language of astronomers, having the geocentric place of 
a planet, it is required to find its heliocentric place. 

369. The first steps in this process are the same as in the 
case of the sun and moon. That is, for the purpose of finding 
the right ascension and declination, the planet is observed on 
the meridian with the Transit Instrument and Mural Circle 
(see Arts. 155 and 230), and from these observations, the 
planet's geocentric longitude and latitude are computed by 
spherical trigonometry. The distance of the planet from the 
sun is known nearly by Kepler's law. From these data it is 
required to find the heliocentric longitude and latitude. 

Let S and E (Fig. 71) be the sun and earth, ASOEH the 



ELEMENTS OF THE PLANETARY ORBITS. 227 

plane of the ecliptic, SA and EH parallels in that plane, point- 
ing to the vernal equinox (which is considered infinitely dis- 
tant), P the planet, PO the perpendicular from it to the plane 
of the ecliptic. Let HEO, the angular distance in the plane 

Fig. 71. 




of the ecliptic (from H to O on the right side of the lines), be 
the geocentric longitude of the planet ; then ASO will be its 
heliocentric longitude. Also PEO, the angular distance of the 
planet from the ecliptic in a plane perpendicular to it, is the 
geocentric latitude, and PSO is the heliocentric latitude. The 
planet's angular distance from the sun, PES, is also known 
from observation. Hence, in the triangle SEP, we know SP 
and SE and the angle SEP, from which we can find PE ; and 
knowing PE and the angle PEO, in the right-angled triangle, 
we calculate EO. Next, in the triangle OES, EO and ES are 
known ; also OES, found by subtracting the planet's longitude 
from the sun's, i. e., HEO from HES (reckoned to the right 
from H) ; hence OSE and OS are calculated. If OSE be added 
to ASE, the supplement of HES, we have ASO, the heliocen- 
tric longitude of the planet. The line OS just found, is called 
the curtate distance of the planet from the sun ; and this, with 
the actual distance SP, gives us, in the right-angled triangle 
SPO, the heliocentric latitude PSO. Thus, by a few processes 
in plane trigonometry, we can do what is equivalent to making 
a transfer of our position from the earth to the sun. 

370. Having now learned how observations made at the 
earth may be converted into corresponding observations made 



228 THE PLANETS. 

at the sun, we may proceed to explain the mode of finding the 
several elements before enumerated ; although our limits will 
not permit us to enter further into detail on this subject, than 
to explain the leading principles on which each of these ele- 
ments is determined.* 

37 1. First, to determine the position of the Nodes, and the 
Inclination of the Orbit. 

These two elements, which deter- Fi s- 72 - 

mine the situation of the orbit (Art. J?. ( 

867), may be derived from two helio- 
centric longitudes and latitudes. Let a. jf 
ANRS be an arc of the ecliptic, A 

the vernal equinox, NQ an arc of a planet's orbit, and IT its 
node. Let AR, AS, be the two heliocentric longitudes ; PR, 
QS, the corresponding heliocentric latitudes, which have been 
determined as by Art. 369. 

By Napier's rule, 

sin NS = tan QS x cot PNR, 
and sin NR = tan PR x cot PNR. 

Eliminating cot PNR, we have 

smNS_sin_NR # 

tan QS~tanPR' 
substituting AS - AN for NS, and AR - AN for NR, then 

sin AS x cos AN — cos AS x sin AN _ 
tan QS _ 

sin AR x cos AN — cos AR x sin AN 
tan PR ' ' 

.*. (cos AR x tan QS — cos AS x tan PR) sin AN = 
(sin AR x tan QS — sin AS x tan PR) cos AN. 

m sin AN , a AT\ — sm ^^ x tan Q^ ~" s * n ^ x tan "^ 

' " cos AN ' ~~ cos AR x tan QS - cos AS x tan PR' 

Thus AN, the longitude of the node is found ; for all the 
quantities on the second member of the equation are known. 
Again, since AN is found, we may deduce from the first 

* Most of these elements admit of being determined in several different ways, 
an explanation of which may be found in the larger works on Astronomy. 



ELEMENTS OF THE PLANETARY ORBITS. 229 

equation above, the value of PNR, which is the inclination of 
the orbit. 

372. Secondly, to find the Periodic Time. 

This element is learned, by marking the interval that passes 
from the time when a planet is in one of the nodes until it re- 
turns to the same node. We may know when a planet is at 
the node, because then its latitude is nothing. If, from a series 
of observations on the right ascension and declination of a 
planet, we deduce the latitudes, and find that one of the obser- 
vations gives the latitude 0, Ave infer that the planet was at 
that moment at the node. But if, as commonly happens, no 
observation gives exactly 0, then we take two latitudes that 
are nearest to 0, but on opposite sides of the ecliptic, one south 
and the other north, and as the sum of the arcs of latitude is 
to the whole interval, so is one of the arcs to the corresponding 
time in which it was described, which time being added to the 
first observation, or subtracted from the second, will give the 
precise moment when the planet was at the node. 

By repeated observations it is found, that the nodes of the 
planets have a very slow retrograde motion. 

373. If the orbit of a planet cut the ecliptic at right angles, 
then small changes of place would be attended by appreciable 
differences of latitude ; but in fact the planetary orbits are in 
general but little inclined to the ecliptic, and some of them lie 
almost in*the same plane with ft. Hence arises a difficulty in 
ascertaining the exact time when a planet reaches its node. 
Among the most valuable observations for determining the 
elements of a planet's orbit, are those made when a superior 
planet is in or near its opposition to the sun, for then the helio- 
centric and geocentric longitudes are the same. When a num- 
ber of oppositions are observed, the planet's motion in longi- 
tude, as would be observed from the sun, will be known. The 
inferior planets also, when in superior conjunction, have their 
geocentric and heliocentric longitudes the same. When in in- 
ferior conjunction, these longitudes differ 180°; but the inferior 
planets can seldom be observed in superior conjunction, on 
account of their proximity to the sun, nor in inferior conjunc- 
tion except in their transits, which occur too rarely to admit of 



230 THE PLANETS. 

observations sufficiently numerous. Therefore, we can not so 
readily ascertain by simple observation, the motions of the 
inferior planets seen from the sun, as we can those of the su- 
perior.* 

374. Hence, to obtain accurately the periodic time of a su- 
perior planet, we find the interval elapsed between two opposi- 
tions separated by a long interval, when the planet was nearly 
in the same part of the zodiac. From the periodic time, as 
determined approximately by other methods, it may be found 
when the planet has the same heliocentric longitude as at the 
first observation. Thus the time of a complete number of rev- 
olutions will be known, and thence the time of one revolution. 
The greater the interval of time between the two oppositions, 
the more accurately the periodic time will be obtained, be- 
cause the errors of observation will be divided between a great 
number of periods ; therefore by using very accurate observa- 
tions, much precision may be attained. For example, the 
planet Saturn was observed in the year 228 b. c, March 2 
(according to our reckoning of time), to be near a certain star 
called y Yirginis, and it was at the same time nearly in oppo- 
sition to the sun. The same planet was again observed in op- 
position to the sun, and having nearly the same longitude, in 
Feb., 1714. The exact difference between these dates was 
1943y. 118d. 21h. 15m. It is known from other sources, that 
the time of a revolution is 29J years nearly, and hence it was 
found that in the above period there were 66 revolutions of 
Saturn ; and dividing the interval by this number, we obtain 
29.444 years, which is nearly the periodic time of Saturn ac- 
cording to the most accurate determination. 

375. Thirdly, to determine the distance from the sun, and 
major axes of the planetary orbits. 

The distance of the earth from the sun being known, the 
mean distance of any planet (its periodic time being known) 
may be found by Kepler's law, that the squares of the periodic 
times are as the cubes of the distances. The method of find- 
ing the distance of an inferior planet from the sun by observa- 

« Brinkley, p. 167. 



ELEMENTS OF THE PLANETARY OEBITS. 231 

tions at the greatest elongation, has been already explained 
(see Art. 308). The distance of a superior planet may be 
found from observations on its retrograde motion at the time 
of opposition. The periodic times of two planets being 
known, we of course know their mean angular velocities, 
which are inversely as the times. Therefore, let Ee (Fig. 73) 
be a very small portion of the earth's orbit, and Mm a corre- 
sponding portion of that of a superior planet, described on the 

Fisr. To 



day of opposition, about the sun S, on which day the three 
bodies lie in one straight line SEMX. Then the angles ES<? 
and MSm, representing the respective angular velocities of the 
two bodies, are known. Now if em be joined, and prolonged 
to meet SM continued in X, the angle EXe, which is equal to 
the alternate angle ~K.ey, being equal to the retrogradation of 
the planet in the same time (being known from observation), 
is also given. E#, therefore, and the angle EX^ being given 
in the right-angled triangle EXe, the side EX is easily calcu- 
lated, and thus SX becomes known. Consequently, in the 
triangle SmX, we have given the side SX, and the two angles 
mSX and wsXS, whence the other sides Sm and mX are easily 
determined. Now Sm is the radius vector of the orbit of the 
superior planet at the point through which it was passing at 
the time of the observation- Of course, one such observation 
could not be relied on as giving the mean distance; but it 
would be a satisfactory approximation in the case of any plan- 
etary orbit, since these orbits are all very nearly circular. 
And by repeating the process every year, as the earth passes 
between the sun and planet, the average of all will ultimately 
express the mean distance, or semi-major axis of the orbit in 
question.* 

376. Fourthly, to determine the place of the perihelion, 
the time of passing it, and the eccentricity. 

A method applicable to the inferior planets, is to make a 

* Sir J. Herschel. 



232 



THE PLANETS. 



series of observations upon them at the times of their greatest 
elongations, as described in Art. 308. If the length of the 
radius vector be thus obtained at each return to the point of 
greatest elongation, there will be found among them all, one 
that is a maximum, and another that is a minimum. The 
latter point is approximately the place of the perihelion. Thus 
(Fig. 60), if in a long series of observations on the greatest 
elongations of Mercury, the value of SB were, at a certain 
time, to be the least of all, we should know that that point is 
the place of perihelion, and, of course, that the point diametri- 
cally opposite is the place of the aphelion. Moreover, by cal- 
culating the distances of the planet from the sun at these two 
points, as described in Art. 308, we ascertain the length of the 
least and greatest radius vector; and half the difference of 
these two lines constitutes the eccentricity. 

For the superior planets, we might suppose the method de- 
scribed in Art. 375, to be pursued every year at the time of 
opposition, till a radius vector is found, which is less than any 
other; that is approximately the place of perihelion. For 
the most distant planets, however, the angle of retrogradation 
X^y, or <?XS, is so minute, that the calculated value of Sm is 
very uncertain. Moreover, the distant planets pass their 
aphelion and perihelion at long intervals. So that, in a given 
case, it may be necessary to wait 40 years for Uranus, or 80 
for J^eptune, to pass either of the apsides. 

But trigonometry, building on a 
few instrumental observations, af- 
fords other modes of arriving at these 
elements of a planetary orbit, one 
of which is derived from the greatest 
equation of the center (Art. 200). For 
since the two points in the orbit 
where this becomes greatest are 
equally distant from the apsides, by 
bisecting the interval between these 
two points, we obtain the position of 
the perihelion and aphelion. Let 
AEBF (Fig. 74) be the orbit of the 
planet, having the sun in the focus 
at S. In an ellipse, the square root of the product of the semi- 



Fisr. 74. 




ELEMENTS OF THE PLANETARY ORBITS. 233 

axes gives the radius of a circle of the same area as the 
ellipse. Therefore, with the center S, at the distance SE= 
VAKxOK, describe the circle CEGF, then will the area of 
this circle be equal to that of the ellipse. At the same time 
that a body departs from A the aphelion, let a body begin to 
move with a uniform motion from C through the periphery 
CEGF, and perform a whole revolution in the same period 
that the planet describes the ellipse ; the motion of this body 
will represent the equable or mean motion of the planet, and 
it will describe around S areas or sectors of circles which are 
proportional to the times, and equal to the elliptic areas de- 
scribed in the same time by the planet. Suppose the body 
describing the circle to be at M ; then taking the sector ASP 
= CSM, P is the true place of the planet. The angle CSM is 
is the mean anomaly,* CSD the true, and DSM the equation 
of the center. But in a circle, sectors vary as their arcs ; 
therefore, the sector DSM may be used for the equation ; but 
taking CSD from the equals CSM and ASP, DSM=ACDP, 
which therefore measures the equation of the center. Now 
ACDP obviously increases till it becomes ACE, that is, when 
the planet has reached the point where the two orbits intersect. 
It may be shown, that after passing E, the equation dimin- 
ishes. The half orbits AEB, CEG, are described in the same 
time; and the mean place, therefore, remains in advance of 
the true, till they reach G and B together. Let Y be the 
mean place, and R the true place, at a certain moment ; then 
the angle CSY is the mean anomaly, and CSm the true, and 
YSm the equation. The sectors ASR, CSY, are equal, being 
described in the same time. Taking CERS from each, ACE = 
EraK+YSra; .-. YSra=ACE-EmR; that is, the equation 
YSm has diminished by EmR since the planet was at E. 
Therefore E is the place of greatest equation of the center. 

But E is also the place where the mean and true angular 
motions are equal, because the equal sectors of the two orbits 
described in each instant, have the same length at that point, 
and therefore the same angle. Hence, the greatest equation 
occurs where the mean angular motion is equal to the true. 

* Anomaly is now reckoned from perihelion (Art. 200) ; but that change does 
not affect the correctness of this reasoning. 



234: THE PLANETS. 

The mean angular motion being known from the periodic time 
of the planet, it is then ascertained by observation when the 
true motion equals it, and thus the time of the greatest equa- 
tion of the center is obtained. Now, this occurs twice in the 
revolution, at E and F ; and half way between these points lie 
the apsides A and B. Therefore, observing the times of the 
greatest equation of the center, E and F, and bisecting the in - 
terval, we have the time of the planet's passing the perihelion 
B. But the same observations also determine the heliocentric 
places of E and F, and the middle of the arc EBF is the place 
of the perihelion. 

37 7. The amount of the greatest equation evidently de- 
pends on the eccentricity of the orbit, since it arises wholly 
from the departure of the ellipse from the figure of a perfect 
circle ; hence, the greatest equation affords the means of deter- 
mining the eccentricity itself. In orbits of small eccentricity, 
as is the case with most of the planetary orbits, it is found that 
the arc which measures the greatest equation is very nearly 
equal to the distance between the foci, which always equals 
twice the eccentricity, the measure of the eccentricity being 
the distance from the focus to the center of the ellipse. The 
angular value of radius is 57° 17 ; 44". 8 ; for, 

3.14159 : 1 : : 180° : 57° 17' 44 // .8. 

Therefore, 57° 11' 44". 8 : radius : : half the greatest equation 
of the center : the eccentricity* 

The foregoing explanations of the methods of finding the 
elements of the orbits, will serve in general to show the learner 
how these particulars are or may be ascertained: yet the 
methods actually employed are usually more refined and intri- 
cate than these. In astronomy, scarcely an element is presented 
simple and unmixed with others. Its value, when first disen- 
gaged, must partake of the uncertainty to which the other ele- 
ments are subject, and can be supposed to be settled to a 
tolerable degree of correctness, only after multiplied observa- 
tions and many revisions. f Indeed, a large part of the most 
arduous labors of astronomers have been employed in finding 

* Vince's Complete System, i., p. 113. f Woodhouse, p. 579. 



QUANTITY OF MATTER IN THE SUN AND PLANETS. 235 

the elements of the planetary orbits, with the wonderful degree 
of precision which has finally been attained. 



QUANTITY OF MATTER IN THE SUN AND PLANETS. 

378. It would seem at first view very improbable, that an 
inhabitant of this earth would be able to weigh the sun and 
planets, and estimate the exact quantity of matter which they 
severally contain. But the principles of Universal Gravitation 
conduct us to this result, by a process remarkable for its sim- 
plicity. By comparing the relations of a few elements that are 
known to us, we ascend to the knowledge of such as appeared 
beyond the pale of human investigation. We learn the quantity 
of matter in a body by the force of gravity it exerts. Let us see 
how this force is ascertained. 

379. The quantities of matter in two bodies may be found 
in terms of the distances and periodic times of two bodies re- 
volving around them respectively, being as the cubes of the dis- 
tances divided by the squares of the periodic times. 

The force of gravity G in a body whose quantity of matter 
is M and distance D, varies directly as the quantity of matter, 

M 

and inversely as the square of the distance ; that is, G <x> =- 2 . 

But it is shown of circular orbits (Art. 177), that the force of 
gravity also varies as the distance divided by the square of the 

periodic time, or G oo -=^. Therefore, ^ °° p~2> an( ^ -^ °° p2- 

Thus we may find the respective quantities of matter in the 
earth and the sun by comparing the distance and periodic time 
of the moon revolving around the earth, with the distance and 
periodic time of the earth revolving around the sun. For the 
cube of the moon's distance from the earth divided by the 
square of her periodic time, is to the cube of the earth's dis- 
tance from the sun divided by the square of her periodic time, 
as the quantity of matter in the earth is to that in the sun. 
„ . 238,54:5 s 95,000,000 s , QKO QQ „ _ . . 

That 1S ' "vfW : nkaw ::1: 3o3 ' 8bo - The raost exact 

determination of this ratio gives for the mass of the sun 



9,2* 



THE PLANETS. 



354,936 times that of the earth. Hence it appears that the sun 
contains more than three hundred and fifty-four thousand times 
as much matter as the earth. Indeed, the sun contains eight 
hundred times as much matter as all the planets. 

Another method, well suited to popular illustration, of weigh- 
ing the earth against the sun, is the following. Knowing the 
radii of the solar and lunar orbits respectively, we can easily 
find the space which the moon descends toward the earth, and 
the earth toward the sun, in any given time, as an hour, 
Thus (Fig. 75), if we know the radius AE of the orbit, we can 




determine the length of the arc A5, described in an hour, and 
also the length of the hypotenuse BE. But BE— AE=B&, 
the space through which the central attracts the revolving body 
in the given time. It was shown (Art. 182) that the earth 
draws the moon from a tangent, .0536 of an inch in a second : 
if a calculation of the same kind be made in relation to the 
orbit of the earth, it will be found that the sun draws the earth 
nearly .12 of an inch per second from a tangent ; that is, the 
sun exerts a force 2J greater on the earth than the earth does 
on the moon. But were the sun at the same distance as the 
moon, his force of attraction would be the square of 400, or 
160,000 times as great as it is now; that is, it would be 
2Jx 160,000 times as great as the earth's attraction, and, con- 
sequently, must have 2^x160,000 = 352,000 times as much 
matter — a result agreeing nearly with the former. The agree- 
ment would be exact if more precise numbers were employed, 
but our object is here merely to illustrate the method. 



QUANTITY OF MATTER IN THE SUN AND PLANETS. 237 

380. The mass of each of the other planets that have satel- 
lites may be found by comparing the periodic time of one of 
its satellites with its own periodic time around the sun. By 
this means we learn the ratio of its quantity of matter to that 
of the sun. The masses of those planets which have no satel- 
lites, as Yenus or Mars, have been determined by estimating 
the force of attraction which they exert in disturbing the mo- 
tions of other bodies. Thus, the effect of the moon in raising 
the tides, leads to a knowledge of the quantity of matter in 
the moon ; and the effect of Yenus in disturbing the motions 
of the earth, indicates her quantity of matter.* 

381. The quantity of matter in bodies varies as their mag- 
nitudes and densities conjointly. Hence, their densities vary 
as their masses divided by their magnitudes ; and since we 
know the magnitudes of the planets, and can compute as above 
their masses, we can thus learn their densities, which, when 
reduced to a common standard, give us their specific gravities, 
or show us how much heavier they are than water. "Worlds, 
therefore, are weighed with the same certainty as a pebble, or 
an article of merchandise. 

The densities and specific gravities of the sun, moon, and 
planets, are estimated as follows :f 

Density. Specific Gravity. 

Sun, 0.25 1.37J 

Moon, . 0.56 3.07 

Mercury, 1.12 6.13 

Yenus, , 0.92 5.04 

Earth, . 1.00 5.48 

Mars, 0.95 5.20 

Jupiter, 0.24 1.31 

Saturn, 0.14 0.76 

Uranus, 0.24 1.31 

Neptune, ...... 0.14 0.76 

° These estimates are made by the most profound investigations in Laplace's 
Mecanique Ce'leste, vol. iii. 

f Herschel. 

% The earth being taken, according to Baily, at 5.48, the specific gravities of 
the other bodies (which are found by multiplying the density of each by the 
specific gravity of the earth) are here stated somewhat higher than they are 
given in most works. 



238 THE PLANETS. 

From tins table it appears that the sun consists of matter 
but little heavier than water ; but that the moon is more than 
three times as heavy as water, though less dense than the 
earth, which is five and a half times heavier than water. It 
also appears that the nearer planets are more dense than the 
more remote. Mercury is heavier than most metallic ores, 
while Saturn and Neptune are one-fourth lighter than water. 
The density, however, does not, in all cases, diminish outward ; 
for Yenus is less dense than the earth, and Saturn than 
Uranus. 



CHAPTEE XII. 

PERTURBATIONS OF THE PLANETS STABILITY OF THE SYSTEM 

NUMERICAL RELATIONS OF THE PLANETS PROBLEMS. 

382. The perturbations occasioned in the motions of the 
planets by their action on each other are very numerous, since 
every body in the system exerts an attraction on every other, 
in conformity with the law of universal gravitation. Yenus 
and Mars, approaching as they do at times comparatively near 
to the earth, sensibly disturb its motions; and Jupiter and 
Saturn, although very far asunder, still, in consequence of 
their great masses, exert on each other, when on the same side 
of the heavens especially, a decided influence. Moreover, the 
sun, by his unequal action on the several planets, in conse- 
quence of the peculiar figure of each, produces various irregu- 
larities in their motions. As in the case of the earth and 
moon (Art. 243), these perturbations are divided into periodi- 
cal and secular : periodical, when completed in comparatively 
short periods, as those, for example, which undergo all their 
changes during one revolution of the planet ; and secular, 
when completed only in very long periods, as those which 
affect the form and inclination of the orbits. 

383. If the only bodies in the system were a central body 
like the sun, and a revolving body like Yenus, then, when the 



PERTURBATIONS OF THE PLANETS. 239 

planet was once put in motion with such a projectile force as 
to make it describe an ellipse, it would forever continue to 
describe the same figure without the least variation, the radius 
vector always passing over equal spaces in equal times ; but 
now introduce a third body so near as to exert on it a decided 
attraction, and its motions no longer retain their simplicity, 
but become complicated by the conflicting influences of the 
two attracting bodies. The sun, however, in consequence of 
its mass, which is eight hundred times as great as that of all 
the planets, and, of course, vastly greater than that of any one 
of them, exerts a force so much superior to that of any or all 
the other disturbing bodies, that the elliptical figure of the 
orbits is nearly maintained, and a near approximation to the 
place of a planet is obtained by neglecting all those minor 
forces, and simply contemplating it as revolving in an ellipti- 
cal orbit. Still it is essential, in order to find the exact place 
of a planet at any given time, that all these irregularities, 
minute as they may be, be carefully summed up, and their 
resultant applied to the elliptical motions. To investigate 
these perturbations, to estimate their precise amount, and to 
register them in tables, for the use of the practical astronomer, 
have constituted a large part of the labors of modern astrono- 
my. The knowledge gained by astronomers of the planetary 
motions, considering the very numerous irregularities, both 
periodical and secular, to which they are subject, is truly won- 
derful. The motion of Jupiter, for instance, is so perfectly 
calculated, that astronomers have computed ten years before- 
hand the time at which it will pass the meridian of different 
places, and we find the prediction correct within half a second 
of tiine.* The more obvious irregularities have been detected 
by observation ; the more minute, by following out the conse- 
quences of universal gravitation. Even those at first revealed 
to the instruments of the astronomer have been confirmed and 
estimated with greater accuracy by the same far-reaching 
principle ; and many of the irregularities have been first 
Drought to light by this theory, which had before eluded ob- 
servation ; although, when once pointed out as a result of the 
principle of gravitation, careful instrumental measurements 



Airy. 



240 THE PLANETS. 

have confirmed them, except in cases where the force was too 
minute to be reached by the most refined observation. Peri- 
odical perturbations among the bodies of the solar system 
may be compared to the regular flux and reflux of the tides, 
by which the ocean daily oscillates about its mean level, 
without any permanent change of level ; while secular pertur- 
bations would resemble any slow changes of level, which, ac- 
cumulating from time to time, might finally become obvious 
to measures of the depths of the ocean, as recorded from age 
to age. As an example of the extreme minuteness of some of 
these secular perturbations, we may instance the changes in 
the eccentricity of the earth's orbit. The entire eccentricity is 
so small that the figure, when drawn on paper in just propor- 
tions, can scarcely be distinguished from a circle, the focus of 
the ellipse being distant from the center only about ^ part of 
the semi-major axis. But the change of eccentricity in a 
century is only the twenty- five thousandth part of the whole, 
or the fifteen hundred thousandth part of the semi-major axis. 

384. But although the secular inequalities of the planetary 
motions are exceedingly slow, yet may they not, in time, accu- 
mulate so as to derange the whole system ; and do they not, 
at least, indicate that the system carries within it the seeds of 
its own dissolution ? So far is this from being the case, that 
the stability of the solar system is a fact established on the 
most satisfactory evidence, and its demonstration is among the 
finest triumphs of physical astronomy. Even a superficial 
view of the system will convince us that care has been be- 
stowed on this point by several obvious arrangements. One 
is, that the planets have, severally, so small masses, compared 
with the sun, as to interfere but little, at most, with the 
supremacy of his control over the planetary motions. An- 
other is, that the planets are placed at such great distances 
from each other — a distance which * is greater among the 
largest bodies, as Jupiter and Saturn, than among the smaller, 
as the Earth and Yenus ; and another still, that the orbits are 
less eccentric when the masses of the bodies are greater, by 
which provision they are always maintained at a remote dis- 
tance from the sun. Were the orbit of Jupiter as eccentric as 
that of Mars, he would approach so near the earth at his peri- 



STABILITY OF THE SYSTEM. 24:1 

helion, as greatly to endanger its stability. But if even these 
general considerations might convince us that the stability of 
the solar system is provided for, a more profound investiga- 
tion will reveal this truth in a far more admirable light. This 
object is especially secured by the following remarkable pro- 
visions. 

First, by the invariability of the major axes, and of the 
periodic times / secondly, by the fact, that whatever irregu- 
larities a planet undergoes on one side of its orbit (so far as 
respects the periodical perturbations), they are compensated 
on the other side ; so that, when it returns to a given point, 
as the node or the perihelion, any irregularities it may have 
felt in different parts of its orbit neutralize one another, and 
therefore do not constitute an accumulating mass of errors ; 
and, thirdly, by this, that all the secular perturbations are 
restricted within narrow limits, oscillating to and fro ; but, 
before they can proceed so far on one side as to endanger the 
stability of the system, they turn about and proceed, for a 
similar period, in the opposite direction. 

385. These truths have been established by the most rigor- 
ous mathematical demonstrations, by the successive labors of 
three very celebrated mathematicians — Euler, Lagrange, and 
Laplace. It was demonstrated that the major axes of the 
planetary orbits, and the times of their revolutions around the 
sun, are subject to no secular perturbations, nor to any varia- 
tion whatever, but such as, in the course of a single revolution, 
exactly compensate and neutralize each other. This is a most 
important point in relation to the stability of the system ; for 
if the lengths of the major axes varied, then, of course, the 
times of revolution would vary (since, by Kepler's third law, 
the squares of the periodic times are in a constant ratio to the 
cubes of the major axes), and we should have years of unequal 
length, and the earth, by approaching at one time nearer to 
the sun, and at another receding further from it, would render 
the changes of temperature too great for the existence of ani- 
mal or vegetable life ; and similar evils, it is probable, would 
result to the economy of the other planets. It was next estab- 
lished, that the eccentricities of the planetary orbits, although 
they have been undergoing constant changes in all time past, 



242 THE PLANETS. 

and will continue to undergo them in all future ages, can never 
vary beyond a certain moderate limit, entirely within the 
bounds of safety to the stability of the system. The eccentri- 
city of the earth's orbit, for example, has been diminishing ever 
since the creation of man ; and although, as we have seen, the 
rate of diminution is exceedingly slow, yet, in the progress of 
centuries, it would totally change the character of the earth's 
orbit ; first reducing it to the circular form, and finally carry- 
ing its eccentricity to a fatal extreme. In like manner, the 
inclination of the earth's orbit to the equator is constantly di- 
minishing, and is now about two-fifths of a degree less than it 
was in the days of Aristotle ; and, were this to proceed in the 
same direction, the equator and ecliptic would coincide, the 
change of seasons would cease, and the whole economy of na- 
ture would be subverted. But Laplace has demonstrated, that 
such an event can never occur, nor can the entire extent of 
this variation exceed three degrees. It is worthy of remark, 
that those perturbations, such as changes in the place of the 
perihelion, affecting a change of direction in space of the major 
axis of the orbit, or in the place of the nodes, which, by accu- 
mulating, do not endanger the stability of the system, proceed 
onward through the entire circuit of the heavens ; while per- 
turbations which, by indefinite accumulation, would bring ruin 
to the system, such as variations of eccentricity and of inclina- 
tion, are not progressive, but oscillatory, waving to and fro 
within the limits of entire safety. 

386. These great ends would not have been secured, had 
the system been constructed differently from what it is. Nu- 
merous conditions must concur in order to produce these re- 
sults : the mass of the sun must have greatly exceeded that of 
any or all the planets ; the eccentricities of the orbits must have 
been small ; and the planets must all have revolved around the 
sun in the same direction, and in planes but little inclined to each 
other.* It was also necessary that the periodic times of the 
planets should, in general, be incommensurable ; for were 
their periods such that one planet would revolve a certain 



* Laplace, Sys. du Monde. Herschel's outlines. Grant's History of Physical 
Astronomy. Pontecoulant's Trait. Elemen. de Phys. C61este. 



RELATIONS BETWEEN BODIES OF THE SOLAR SYSTEM. 243 

number of times exactly, while another planet, next to it, re- 
volved a certain other even number of times, then, when they 
once came into the sphere of each other's influence, they might 
remain under it so long, and return to their relative position 
so often, as seriously to derange their orbits. An instance of 
this, in fact, occurs in the case of Jupiter and Saturn, five rev- 
olutions of Jupiter being nearly equal to two of Saturn, a re- 
lation which gives rise to what is called the long inequality of 
Saturn and Jupiter. Similar effects result from a near com- 
mensurability of the mean motions of any other two planets. 
One exists between the Earth and Yenus, 13 times the period 
of Yenus being very nearly equal to 8 times that of the Earth ; 
still, the influence of this disturbing cause is so nicely compen- 
sated, and its effects so distributed, that, according to Mr. Airy 
(who was the first to detect it), it amounts, at its maximum, to 
no more than a few seconds for a period of 240 years. The 
laws which regulate the eccentricities and inclinations of the 
planetary orbits (says an able writer on Physical Astronomy), 
combined with the invariability of the mean distances, secure 
the permanence of the solar system throughout an indefinite 
lapse of ages, and offer to us an impressive indication of the 
Supreme Intelligence which presides over nature, and perpet- 
uates her beneficent arrangements. When contemplated 
merely as speculative truths, they are unquestionably the most 
important which the transcendental analysis has disclosed to 
the researches of the geometer ; and their complete establish- 
ment would suffice to immortalize the names of Lagrange and 
Laplace, even although these great geniuses possessed no other 
claim to the recollection of posterity.* 

NUMERICAL RELATIONS BETWEEN THE BODIES OF THE 
SOLAR SYSTKM.f 

387. If we contemplate the relations subsisting between a 
central body, as the sun, and a revolving body, as one of the 

* Grant's Hist. Phys. Ast., p. 56. 

f In the preparation of this article, the author has derived much assistance 
from a small work, now nearly out of print, containing the suhstance of three 
lectures delivered to the students of Yale College in 1781, by Rev. Nehemiah 
Strong, at that time Professor of Mathematics and Natural Philosophy. 



244 THE PLANETS. 

planets, it will be readily understood, that if the quantity of 
matter in the central body is increased, while the distance of 
the revolving body remains the same, the velocity of the re- 
volving body must be increased also, in order to generate a 
sufficient centrifugal force to counterbalance the increased 
force of attraction in the central body, arising from the increase 
of its mass ; and that, were the force of attraction diminished, 
by removing the body to a greater distance from the center, 
then the rate of its motion would also have to be diminished , 
otherwise the centrifugal force would overpower the force of 
attraction. It is a remarkable fact, that the members of the 
solar system are so adjusted to each other, in respect to their 
velocities, distances from the sun, periodic times, and gravita- 
tion toward the central body, that if any one of these particu- 
lars is known, all the rest become known also. Thus, if it 
were found that a new-discovered planet moved with one-sixth 
the velocity of the earth, we should know at once that its dis- 
tance from the sun was thirty-six times as great as the earth's 
distance, that its time of revolution was two hundred and six- 
teen years, and that its gravitation toward the sun was twelve 
hundred and ninety-six times less than that of the earth ; for 
the distance is as the square of the number expressing the recipro- 
cal of its velocity, compared with that of the earth ; its periodic 
time as the cube / and the reciprocal of gravity as the fourth 
power of the same number. All this follows from Kepler's 
third law — that the squares of the periodic times are as the 
cubes of the distances ; and from the law of universal gravita- 
tion — that the force of attraction is inversely as the square of 
the distance. The four particulars named, therefore, constitute 
a series of numbers in geometrical progression, of which the 
first term is equal to the ratio. The truth of this proposition 
may be demonstrated as follows : 

Let D be the mean distance of a planet from the sun, 7r the 
ratio of the diameter to the circumference of a circle, and P the 
time of revolution around the sun, or periodic time ; then the 

expression lor the velocity is V =-w- °° p- And V 2 oo ^. 
But, by Kepler's law, P 2 oo D 3 ; .\ V 2 oo ?? or Y 2 oo A Since 



NUMERICAL RELATIONS. 245 

a body moves with less velocity, when the distance from the 
sun is greater, it will be convenient, in order to avoid fraction- 
al forms, to use the reciprocal of Y, instead of Y itself. Let 

R (retardation) be equal to -^ ; then Y= ^, and Y 2 = — : hence 

— gc =- ; .-. H 2 co D (IV If, therefore, R indicates how much 
Iv L) 

slower a planet moves than another, as the earth, taken as a 
standard, the square of R will show how much further from 
the sun the planet is than the earth. 

B D 3 

Again, since Yco p, Y 3 go ^ . But, by Kepler's law, D 3 co P 2 ; 

«*• v 3 °° p ? ° r v 3 °° p ; •*■ R 3 °° ? (2). 

Consequently, if R expresses how many times slower than 
the earth a given planet moves, the cube of R will express the 
relative periodic time. 

Fftially, by the law of gravitation, the force of gravitation 
toward the central body varies as the square of the distance 

inversely, or G- co =- •. But if the reciprocal of gravity be call- 
ed Levity, and expressed by L, then L co B 2 ; but R 2 co D, 
in R 4 co D 2 , and R 4 co L (3). 

Therefore, if R denotes how much slower a planet moves in 
its orbit 'than the earth, R 4 will denote how much less the 
same body gravitates toward the central body. Collecting 
these several results, it appears that the reciprocal of velocity 
R, the distance D, the periodic time P, and the reciprocal of 
gravity L, are respectively denoted by the geometrical series^ 
R, R 2 , R 3 , R 4 , in which the first term and the ratio are equal. 

388. A number of very useful and convenient rules, may 
be derived from this numerical relation between the members 
of the solar system ; since, when any one of the four things 
named is given, all the rest may be found from it; and each 
of the four may be found in four different ways when the other 
members of the series are given. This will be obvious from a 
few examples. ' 



246 THE PLANETS. 

I. Given the retardation (R). 

1. Square the retardation for the distance. 

2. Cube the retardation for the periodic time. 

3. Take the fourth power of the retardation for the recipro- 
cal of gravity. 

II. Given the distance (D). 

1. Take its square root for reciprocal of velocity. 

2. Take the cube of the square root of the distance for the 
periodic time. 

3. Take its square for the reciprocal of gravity. 

III. Given the periodic time (P). 

1. Take the cube root of the periodic time for the reciprocal 
of velocity. 

2. Take the square of the cube root of the periodic time for 
the distance. 

3. Take the fourth power of its cube root for the reciprocal 
of gravity. 

IV. Given the reciprocal of gravity (L). 

1. Take the fourth root for the reciprocal of velocity. 

2. Take the square root for the distance. • 

3. Take the cube of the fourth root for the periodic time. 

V. Required the reciprocal of velocity. 

This may be obtained by taking the square root of the dis- 
tance, or the cube root of the periodic time, or the fourth root 
of the reciprocal of gravity, or by dividing the reciprocal of 
gravity by the periodic time. 

YI. Required the distance. 

Take the square of the reciprocal of velocity, or the square 
of the cube root of the time, or the square root of the reciprocal 
of gravity, or divide the time by the reciprocal of velocity. 

VII. Required the periodic time. 

Take the cube of the reciprocal of velocity, or the cube of 
the square root of the distance, or the f power of the reciprocal 
of gravity, or divide the reciprocal of gravity by that of ve- 
locity. 

VIII. Required the diminished gravitation. 

Take the fourth power of the reciprocal of velocity, or the 
square of the distance, or the J power of the time, or multiply 
the time by the reciprocal of velocity. 

According to the foregoing rules, tables may be formed, ex- 



PROBLEMS. 



247 



hibiting, in a striking light, the numerical relations of the 
members of the solar system. In the following table the dis- 
tances are taken from Herschel's Astronomy, and from these 
the other particulars are determined by the preceding rules. 
If Mercury were taken as the standard of comparison, then the 
retardations of all the other planets would be greater than 
unity ; but, as it is convenient to take the earth as the stand- 
ard, the retardations of Mercury and Yenus will be less than 
unity ; showing that the velocity (which is expressed by the 
fraction inverted) is greater than that of the earth. In like 
manner, the force of gravitation of an inferior planet, being 
greater than that of the earth, is the reciprocal of the tabular 
number. 

Table showing the Numerical Relations of the Primary 

Planets. 



Planets- 


Retardations. 


Distances. 


Per. Times. 


Recip. of Gravity. 


Mercury 


0.62217 


0.38710 


0.24084 


0.14985 


Venus 


0.85049 


0.72333 


0.61519 


0.52321 


Earth 


1.00000 


1.00000 


1.00000 


1.00000 


Mars 


1.23440 


1.52369 


1.88080 


2.32170 


Jupiter 


2.28100 


5.20277 


11.86700 


27.06900 


Saturn 


3.08850 


9.53878 


29.46100 


90.98900 


Uranus" 


4.37970 


19.18239 


84.01200 


367.95000 


Neptune 


5.49040 


30.14512 


165.51000 


908.72000 



389. Problems. 

Prob. 1. — The planet Pallas was discovered to have a period 
of about 4f years. How much slower does it move in its orbit 
than the earth — how much further is it from the sun — and 
how much less does it gravitate toward the sun? Ans. R = 
1.67, D = 2.79, L = 7.80. 

By applying the proportional numbers determined by this 
problem respectively to the earth's motion per second, to its 
distance from the sun in miles, and to the space through which 
the earth departs in a second from a tangent to her orbit, we 
may obtain the numerical value of each of these elements. 

Prob. 2. — What would be the periodical time of a meteor or 
planet revolving close to the earth. 



248 THE PLANETS. 

As the moon is a body revolving around the earth at a known 
distance, and with a known periodic time, it will evidently 
furnish the necessary standard of comparison. The distance of 
the moon from the center of the earth being 60 times the 
earth's radius, and, of course, 60 times that of the meteor, its 
rate of motion is \/60 times less. The retardation being n/60, 

.3 

the periodic time will be 60 2 . Now, what part of the moon's 

— — 

period is 60 2 ? Divide the moon's period (27.32 days) by 60 2 . 

and we have for the answer, 1 hour, 24 minutes, 38.88 seconds. 

Prob. 3. — What would be the periodic time of a body re- 
volving about the earth at the distance of 5000 miles from the 
center? Am. lh. 59m. 23.28s. 

Prob. 4. — How much faster must the earth revolve in order 
that bodies on its surface may lose all their gravity ? 

According to problem 2, the period of a body revolving at 
the surface of the earth, is 1.4108 hours ; and since, in a cir- 
cular orbit, the force of gravity and the centrifugal force are 
equal, therefore a body like that contemplated in problem 2, 
is in equilibrium between these two forces ; consequently, such 
a body may be considered as having lost all its gravity, and 
being, by the supposition, close to the earth, we have only to in- 
quire how much its velocity exceeds that of the earth. Now, 24 
divided by 1.4108 gives 17.01 ; which shows that were the 
earth to revolve on its axis about 17 times faster than it does 
at present, the bodies on the surface would lose all their weight ; 
and were the velocity greater than this, the centrifugal force 
would prevail over the centripetal, and the same would fly off 
from the earth in tangents. 

Prob. 5. — Were the moon to be removed so far from the 
earth as to revolve about it but once a year, how much greater 
would be its distance than at present, how much less its veloci- 
ty, and its gravitation toward the earth ? 

Its period being increased 13.37 times, its retardation is 

13.37* = 2.373 ; its distance 2.373 2 = 5.631 ; and its diminished 
gravity 5.631 2 = 31.71. Or R = 2.373, D = 5.631, and L = 
31.71. 

Multiplying the present distance of the moon, 238,545 miles, 
by 5.631, we obtain about 1,343,000 miles for the distance at 



PROBLEMS. 249 

which the moon must have been placed in order to complete 
its revolution in one year. 

Peob. 6. — Were the earth's mass equal to the sun's, and of 
course 354,000 times as great as at present, in what time would 
the moon revolve around it ? 

Since the masses are as the cubes of the distances divided by 
the squares of the periodic times, letting the required time 
be denoted by x, 1 (the earth's mass) : 354.000 (the sun's mass) 
P 3 , D s . , 1 1 1_ 354,000 , 27.32 

: 2T.32 2 *' W : : 2T.32 2 : a? 2 ' '"* ~x~ l ~ 2T.32 2 ' ''' X ~~~ v/354,000 ~ 
lh. 6m. 7s. 

Comets, in passing their perihelion, especially when that 
happens to be very near the sun, as in the great comet of 1843, 
move with an astonishing rapidity ; requiring a velocity not 
merely sufficient to generate the centrifugal force necessary to 
balance the powerful force of attraction exerted by the sun v 
but greatly to exceed that force, since they are carried far 
without a circular orbit into an elliptical or even a hyperbolic 
orbit. 

Peob. 7. — The perihelion distance of the great comet of 1843 
being 502,000 miles from the center of the sun, what must 
have been its velocity per hour, if in a circular orbit ? 

Peob. 8. — How much must the mass of the earth be in- 
creased in order that the moon may revolve about it in the 
same time as at present, when removed to three times her 
present distance ? 

Peob. 9. — How much must the mass of the earth be in- 
creased to make the moon, at her present distance, revolve in 
24 hours ? 

Peob. 10. — The semi-diameter of Jupiter being 11 times 
that of the earth, and the distance of its fourth satellite from 
the center of the planet being 27 times the radius of the 
planet; also the sidereal revolution of the satellite being 16.69 
days, while that of the moon is 27.3217 days, and her distance 
60 times the radius of the earth : How much does the quantity 
of matter in Jupiter exceed that of the earth ? Ans. 324.49 
times. 

Peob. 11. — Suppose volcanic matter to be thrown from the 
moon toward the earth, required the point where it would be 
in equilibrium between the two, the mass of the moon being 



250 THE PLANETS. 

one-eightieth that of the earth ? Ans. 24,000 miles from the 
center of the moon, nearly. 

Prob. 12. — Suppose that the only two bodies in the universe 
were a sphere two inches in diameter, of the same density with 
the earth, for the primary, and a material point for the satel- 
lite. What would be the periodic time of the satellite, at the 
distance of one foot, in a circular orbit ? Ans. 2 days, 10 
hours, 13 minutes.* 

6 The elements used in the solution of this problem are, for the diameter of 
the earth, 7912.4 ; for the distance of the moon 238,545 miles ; and for its peri- 
odic time, 27.32 days. The solution, conducted in the ordinary mode, will be 
found susceptible of great abridgment. But the following ingenious method is 
still shorter. It was suggested to the author by one of his pupils, Mr. Samuel 
Emerson, of the class of 1848. 

Lemma. The periodic times of two satellites revolving about primaries of equal densities, 
at distances which are equimultiples of their radii, are equal. 

Demonstration. Let 

M, m = the masses of the two bodies respectively. 

P, p = the periodic times. 

R, r — the radii of the spheres. 

D, d = the distances of their satellites. 

D 3 ds 
Then, M : m : : — : -. 

But since D and d are equimultiples of R, r, by some number n, therefore 

D 3 =R 3 ?i 3 , and d* = r 3 n s ; 

R 3 /t 3 r 3 n 3 R 3 r 3 
Hence, M : m : : — — : — : : — - : —z. But, R 3 and r 3 oo M and m. 
pa p pa p i 

' M wi MXm MXm 

Therefore, M. : m : : — : — , .\ = , .*. P=z>. 

P 2 p*' p'i P 2 ' r 

The moon being distant 60.296 radii of the earth (as would result from the 
above elements), at the distance of 60.296 inches that of the small satellite from 
its primary would be the same multiple of its radius, and consequently, its peri- 
odic time the same. What then is its period at 12 inches ? 

27. 32 2 :p* : : 60.296 3 : 12 s , .-. jt> = 2d. 10b. 13m. 

Corollary. — If any two spheres of the same density be taken, the periodic times 
of satellites revolving about them close to the surface, will be the same in both ; for 
the case becomes this when n=l. Thus, the material point supposed in the 
above problem, will revolve about its little globe in the same time that the 
moon would revolve about the earth, both being situated close to the surfaces of 
their respective primaries. 



CHAPTEE XIII. 

COMETS METEORIC SHOWERS. 

390. A Comet,* when perfectly formed, consists of three 
parts — the Nucleus, the Envelope, and the Tail. The Nucleus, 
or body of the comet, is generally distinguished by its forming 
a bright point in the center of the head, conveying the idea of 
a solid, or at least of a very dense portion of matter. Though 
it is usually exceedingly small when compared with the other 
parts of the comet, yet it sometimes subtends an angle capable 
of being measured by the telescope. The Envelope (sometimes 
called the coma) is a dense nebulous covering, which fre- 
quently renders the edge of the nucleus so indistinct, that it is 
extremely difficult to ascertain its diameter with any degree of 
precision. Many comets have no nucleus, but present only a 
nebulous mass extremely attenuated on the confines, but grad- 
ually increasing in density toward the center. Indeed, there 
is a regular gradation of comets, from such as are composed 
merely of a gaseous or vapory medium, to those which have a 
well-defined nucleus. In some instances on record, astrono- 
mers have detected with their telescopes small stars through 
the densest part of a comet. The Tail is regarded as an ex- 
pansion or prolongation of the coma ; and presenting, as it 
sometimes does, a train of appalling magnitude, and of a pale, 
portentous light, it confers on this class of bodies their pecu- 
liar celebrity. 

391. The number of comets belonging to the solar system, 
is probably very great. Many, no doubt, escape observation 
by being above the horizon in the daytime. Seneca mentions, 
that during a total eclipse of the sun, which happened 60 years 
before the Christian era, a large and splendid comet suddenly 
made its appearance, being very near the sun. The elements 
of at least 180 comets have been computed, and arranged in a 

* K6[irj, coma, from the bearded appearance of comets. 



252 



COMETS. 



catalogue for future comparison.* Of these, six are particu- 
larly remarkable, viz., the comets of 1680, 1770, and 1843; 
and those which bear the names of Halley, Encke, and Biela. 
The comet of 1680 was distinguished not only for its astonish- 
ing size and splendor, but is remarkable for having been the 
first comet whose elements were determined on the sure basis 
of mathematics, as was done by Sir Isaac Newton, it having 
appeared in his time. The comet of 1770 is memorable for 



Fig. 76. 



Fig. 77. 





COMET OF 1811, 



COMET OF 1680. 



the changes its orbit has undergone by the action of Jupiter, 
and for having approached very near to the earth. The comet 
of 1843 was the most remarkable in its appearance of all that 
have been seen in modern times, having been visible at noon- 
day. Halley's comet (the same which reappeared in 1835) is 
distinguished as that whose return was first successfully pre- 
dicted, and whose orbit was first accurately determined ; and 
Biela's and Encke's comets are well known for their short 



e See a complete catalogue of cornets, whose elements have been determined, 
in the American Almanac for 1847. 



COMETS. 253 

periods of revolution, which subject them frequently to the 
view of astronomers. Biela's comet, at its return in 1846, dis- 
played another remarkable feature — a separation into two dis- 
tinct parts. This strange peculiarity was first seen from the 
Observatory of Tale College, by Messrs. Herrick and Bradley, 
but was first publicly announced from the Observatory at 
Washington. At one time, the distance of one nucleus from 
the other, was estimated at 157,000 miles. 

392. In magnitude and brightness, comets exhibit a great 
diversity. They are sometimes so bright as to be distinctly 
visible in the daytime, even at noon and in the brightest sun- 
shine, as was the case with that of 1813 ; and such was the 
comet seen at Rome a little before the assassination of Julius 
Caesar. The comet of 1680 covered an arc of the heavens of 
97°, and its length was estimated at 123,000,000 miles.* That 
of 1811 had a nucleus of only 428 miles in diameter, but a tail 
132,000,000 miles long.f Had it been coiled round the earth 
like a serpent, it would have reached round more than 5000 
times. Other comets are of exceedingly small dimensions, the 
nucleus being estimated at only 25 miles; and some which are 
destitute of any perceptible nucleus, appear to the largest 
telescopes, even when nearest to us, only as a small speck of 
fog, or as a tuft of down. The majority of these bodies can be 
seen only by the aid of the telescope. 

The same comet, indeed, has often very different aspects, at 
its different returns. Halley's comet in 1305 was described by 
the historians of that age, as cometa horre?idw magnitudinis ; 
in 1156 its tail reached from the horizon to the zenith, and 
inspired such terror, that, by a decree of the Pope of Rome, 
public prayers were offered up at noonday in all the Catholic 
churches to deprecate the wrath of heaven, while in 16S2, its 
tail was only 30° in length, and in 1759 it was visible only to 
the telescope, until after it had passed its perihelion. At its 
recent return in 1835, the greatest length of the tail was about 
12°4 These changes in the appearances of the same comet 
are partly owing to the different positions of the earth with 

* Arago. f Milne's Prize Essay on Comets. 

J But might be seen much longer by indirect vision. (Prof. Joslin, Am. Journ. 
Science, xxxi., p. 328.) 



254 



COMETS. 



respect to them, being sometimes much nearer to them when 
they cross its track than at others ; also one spectator so situ- 
ated as to see the comet at a higher angle of elevation or in a 
purer sky than another, will see the train longer than it appears 
to one less favorably situated ; but the extent of the changes 
are such as indicate also a real change in their magnitude and 
brightness. 

393. The periods of comets in their revolutions around the 
sun, are equally various. Encke's comet, which has the short- 
est known period, completes its revolution in 3J years, or more 
accurately, in 1205.23 days; while that of 1811 is estimated to 
have a period of 3,383 years. * The distances to which different 
comets recede from the sun, are also very various. While 
Encke's comet performs its entire revolution within the orbit 
of Jupiter, Halley's comet recedes from the sun to twice the 



Fig. 78. 




jtr 




distance of Uranus, or nearly 3,600,000,000 miles. Figure 78 
is a representation, in due proportions, of the orbit of this 
comet. Its vast dimensions will be truly conceived of by 
reflecting that the radius of the small circle E of the earth's 
orbit implies a space of nearly 100,000,000 miles ; that, as the 
comet recedes from the sun, it soon reaches the orbit of Jupiter, 
and successively traverses the orbits of Saturn, Uranus, and 
Neptune, reaching its aphelion 600,000,000 miles beyond the 
present boundaries of the planetary system. Some comets, 
indeed, are thought to go to a much greater distance from the 



* Milne. 



COMETS. 255 

sun than this, as that of 1811 must have receded from it more 
than 45,000,000,000 miles, while some even are supposed to 
pass into parabolic or hyperbolic orbits, and never to return. 

394. Comets shine by reflecting the light of the sun. In one 
or two instances they have exhibited distinct phases,* although 
the nebulous matter with which the nucleus is surrounded, 
would commonly prevent such phases from being distinctly 
visible, even when they would otherwise be apparent. More- 
over, certain qualities of polarized light enable the optician to 
decide whether the light of a given body is direct or reflected ; 
and M. Arago, of Paris, by experiments of this kind on the 
light of the comet of 1819, ascertained it to be reflected light, f 
The tail of a comet usually increases very much as it ap- 
proaches the sun ; and frequently does not reach its maximum 
until after the perihelion passage. In receding from the sun 
the tail again contracts, and nearly or quite disappears before 
the body of the comet is entirely out of sight. The tail is fre- 
quently divided into two portions, the central parts, in the 
direction of the axis, being less bright than the marginal parts. 
In 1741, a comet appeared which had six tails, spread out like 
a fan. 

The tails, of comets extend in a direct line from the sun, al- 
though they are usually more or less curved, like a long quill 
or feather, being convex on the side next to the direction in 
which they are moving (Fig. 77) ; a figure which may result 
from the less velocity of the portions most remote from the 
sun. Expansions of the envelope have also been at times 
observed on the side next the sun,:f but these seldom attain 
any considerable length. 

395. The quantity of matter in comets is exceedingly small. 
Their tails consist of matter of such tenuity that the smallest 
stars are visible through them. They can only be regarded 
as great masses of thin vapor, susceptible of being penetrated 
through their whole substance by the sunbeams, and reflecting 
them alike from their interior parts and from their surfaces. 



■ Delambre, t. iii , p. 400. f Franoceur, p. 181. 

J See Dr. Joslin's remarks on Halley's comet, Amer. Journ. Science, xxxi. 



256 COMETS. 

It appears, perhaps, incredible that so thin a substance should 
be visible by reflected light, and some astronomers have held 
that the matter of comets is self-luminous ; but it requires but 
Yevj little light to render an object visible in the night, and a 
light vapor may be visible when illuminated throughout an 
immense stratum, which could not be seen if spread over the 
face of the sky like a thin cloud. The highest clouds that 
float in our atmosphere, must be looked upon as dense and 
massive bodies, compared with the filmy and all but spiritual 
texture of a comet.* The small quantity of matter in comets 
is further proved by the fact that they have sometimes passed 
very near to some of the planets without disturbing their mo- 
tions in any appreciable degree. Thus the comet of 1770, in 
its way to the sun, got entangled among the satellites of Jupi- 
ter, and remained near them four months, yet it did not per- 
ceptibly change their motions. The same comet also came 
very near the earth ; so near, that, had its mass been equal to 
that of the earth, it would have caused the earth to revolve in 
an orbit so much larger than at present, as to have increased 
the length of the year 2h. 47m. \ Yet it produced no sensible 
effect on the length of the year, and therefore its mass, as is 
shown by Laplace, could not have exceeded 5^0 of that of the 
earth, and might have been less than this to any extent. It 
may indeed be asked, what proof we have that comets have 
any matter. The answer is, first, they reflect light ; second, 
though not sufficient to disturb so heavy bodies as planets or 
satellites, yet they are themselves exceedingly disturbed by the 
action of the planets, and in exact conformity with the laws of 
universal gravitation. A delicate compass may be greatly 
agitated by the vicinity of a mass of iron, while the iron is 
not sensibly affected by the attraction of the needle. 

396. By approaching very near to a large planet, a comet 
may have its orbit entirely changed. This fact is strikingly 
exemplified in the history of the comet of 1770. At its appear- 
ance in 1770, its orbit was found to be an ellipse, requiring for 
a complete revolution only 5£ years; and the wonder was, 
that it had not been seen before, since it was a very large and 



® Sir J. Herschel. -j- Laplace. 



COMETS. 257 

bright comet. Astronomers suspected that its path had been 
changed, and that it had been recently compelled to move in 
this short ellipse, by the disturbing force of Jupiter and his 
satellites. The French Institute, therefore, offered a high 
prize for the most complete investigation of the elements of 
this comet, taking into account any circumstances which could 
possibly have produced an alteration in its course. By tracing 
back its movements for some years previous to 1770, it was 
found that, at the beginning of 1767, it had entered considera- 
bly within the sphere of Jupiter's attraction. Calculating the 
amount of this attraction from the known proximity of the two 
bodies, it was found what must have been its orbit previous to 
the time when it became subject to the disturbing action of 
Jupiter. The result showed that it then moved in an ellipse 
of greater extent, having a period of 50 years, and having its 
perihelion instead of its aphelion near Jupiter. It was there- 
fore evident why, as long as it continued to circulate in an 
orbit so far from the center of the system, it was never visible 
from the earth. In January, 1767, Jupiter and the comet 
happened to be very near each other, and as both were moving 
in the same direction, and nearly in the same plane, they 
remained in the neighborhood of each other for several months, 
the planet being between the comet and the sun. The conse- 
quence was, that the comet's orbit was changed into a smaller 
ellipse, in which its revolution was accomplished in 5-|- years. 
But as it was approaching the sun in 1779, it happened again 
to fall in with Jupiter. It was in the month of June that the 
attraction of the planet began to have a sensible effect ; and it 
was not until the month of October following that they were 
finally separated. 

At the time of their nearest approach, in August, Jupiter 
was distant from the comet only T J T of its distance from the 
sun, and exerted an attraction upon it 225 times greater than 
that of the sun. By reason of this powerful attraction, Jupiter 
being further from the sun than the comet, the latter was 
drawn out into a new orbit, which, even at its perihelion, came 
no nearer to the sun than the planet Ceres. In this third or- 
bit, the comet requires about 20 years to accomplish its revo- 
lution ; and being at so great a distance from the earth, it is 
invisiblej and will forever remain so, unless, in the course of 

17 



258 COMETS. 

ages, it shall undergo new perturbations and move again in 
some smaller orbit as before.* 

ORBITS AND MOTIONS OF COMETS. 

397. The planets, as we have seen (with the exception of 
the asteroids, which seem to be an intermediate class of bodies 
between planets and comets), move in orbits which are nearly 
circular, and all very near to the plane of the ecliptic, and all 
move in the same direction from west to east. But the orbits 
of comets are far more eccentric than those of the planets ; 
they are inclined to the ecliptic at various angles, being some- 
times even nearly perpendicular to it; and the motions of 
comets are sometimes retrograde. 

398. The elements of a comet's orbit as usually obtained, 
are five : (1), longitude of the node / (2), inclination to the 
ecliptic ; (3), perihelion distance / (4), longitude of perihelion ; 
(5), time of perihelion passage. 

In comparing these with the elements of a planetary orbit 
(Art. 367), we perceive two to be omitted — the periodic time, 
and the eccentricity j while perihelion distance is substituted 
for mean distance. The reason for this is, that in these orbits, 
no reliance can be placed on the determination of the size and 
form of orbit, or the time of describing it, from observations 
which are limited to a short arc near the perihelion. The 
complete problem is not only extremely difficult, as Newton 
pronounced it, but rather wholly impracticable. It is during 
only a very small portion of their course that they are visible 
from the earth, and the observations made in that short period 
can not afterward be verified on more convenient occasions ; 
whereas in the case of the planets, whose orbits are nearly cir- 
cular, and whose movements may be followed uninterruptedly 
throughout a complete revolution, no such impediments to the 
determination of their orbits occur. There is also some una- 
voidable uncertainty in observations made upon bodies whose 
outlines are so ill-defined. It is not unfrequently the case ioo, 
that comets move in a direction opposite to the order of the 



Milne. 



CEBITS AND MOTIONS OF COMETS. 259 

signs in the zodiac, and sometimes nearly perpendicular to the 
plane of the ecliptic ; so that their apparent course through 
the heavens is rendered extremely complicated, on account of 
the contrary motion of the earth. Since it is possible there 
should be any number of elliptic orbits, whose perihelion dis- 
tances are equal, it is obvious that, in the case of very eccentric 
orbits, the slightest change in the position of the curve near the 
vertex, where alone the comet can be observed, must occasion 
a very sensible difference in the length of the orbit (as will be 
obvious from Fig, 79) ; and therefore, though a small error 




produces no perceptible discrepancy between the observed and 
the calculated course, while the comet remains visible from the 
earth, its effect, when diffused over the whole extent of the 
orbit, may acquire a most material or even a fatal importance. 
On account of these circumstances, it is found exceedingly 
difficult to lay down, even in the rudest manner, the path 
which a comet actually pursues, when gone for years beyond 
the limit of our vision ; and least of all to determine with ac- 
curacy the length of the major axis of the ellipse, so as to de- 
rive from it, by Kepler's third law, the time of its revolution, 
and thus to be able to predict its next perihelion passage. An 
error of only a few seconds may cause a difference of many 
hundred years. In this manner, though Bessel determined the 
revolution of the comet of 1769 to be 2,089 years, it was found 



260 COMETS. 

that an error of no more than 5" in observation, would alter 
the period either to 2,678 years, or to 1,692 years. Some as- 
tronomers, in calculating the orbit of the great comet of 1680, 
have found the length of its greater axis 426 times the earth's 
distance from the sun, and consequently its period 8,792 years; 
while others estimate the greater axis 430 times the earth's dis- 
tance, which alters the period to 8,916 years. Newton and 
Halley, however, judged that this comet accomplished its rev- 
olution in only 570 years. 

399. Disheartened by the difficulty of attaining to any pre- 
cision in the circumstances by which an elliptic orbit is char- 
acterized, and, moreover, taking into account the laborious cal- 
culations necessary for its investigation, astronomers usually 
satisfy themselves with ascertaining the elements of a comet 
on the supposition of its describing a parabola ; and, as this is 
a curve whose axis is infinite, the procedure is greatly simpli- 
fied by leaving entirely out of consideration the periodic revo- 
lution. It is true that a parabola may not represent, with 
mathematical strictness, the course which a comet actually fol- 
lows ; but as a parabola is the intermediate curve between the 
hyperbola and ellipse, it is found that this method, which is so 
much more convenient for computation, also accords sufficient- 
ly with observations, except in cases when the ellipse is a com- 
paratively short one, as that of Encke's comet, for example, 
"When the elements of a comet are determined, Kepler's law of 
areas enables astronomers to find, by computation, the exact 
place of the comet in its orbit at any given time, on the sup- 
position that its path is a parabola ; and comparing this place 
with that determined by observation for the same instant, it is 
seen whether the orbit is truly parabolic, or whether its devia- 
tion from that path is such as to indicate that its real path is an 
ellipse ; and the amount of such deviation will give some idea 
of the degree of eccentricity of the ellipse. 

400. The elements of a comet, with the exception of its pe- 
riodic time, are calculated in a manner similar to those of the 
planets. Three good observations on the right ascension and 
declination of the comet (which are usually found by ascer- 
taining its position with respect to certain stars, whose right 



ORBITS AND MOTIONS OF COMETS. 



261 



ascensions and declinations are accurately known) afford the 
means of calculating these elements. 

The appearances of the same comet at different periods of its 
return are so various (Art. 392), that we can never pronounce 
a given comet to be the same with one that has appeared before, 
from any peculiarities in its physical aspect. The identity of 
a comet with one already on record, is determined by the 
identity of the elements. When a new comet appears, we first 
determine its elements, and then turning to a catalogue of 
comets whose elements have previously been found and placed 
on record, we see whether these new elements agree with any 
set of those in the catalogue. If they do, we infer that the 
present comet is identical with that on record ; and the interval 
between the two appearances of the body will indicate its pe- 
riodic time. If, for example, we find respecting a comet now 
visible in the sky, that its path makes the same angle with the 
ecliptic as that of a certain comet in our catalogue, that it 
crosses the ecliptic in the same degree of longitude, that it 
comes to its perihelion in the same place, that its perihelion 
distance is the same, and its course the same in regard to the 
order of the signs, then we infer that the two bodies are one 
and the same; and the number of years that have elapsed 
since its former appearance indicates the period of its revolu- 
tion around the sun. But if these particulars differ wholly 
from any set of recorded elements, we infer that the present is 
a comet which has never visited our sphere before, or at least 
one whose* elements have not been determined and recorded. 
It was by this, means that Halley first established the identity 
of the comet which bears his name, with one that had appeared 
at several preceding ages of the world, of which so many par- 
ticulars were left on record, as to enable him to calculate the 
elements at each period. These were as in the following table. 



Time of ap- 
pearance. 


Inclination of 
the Orbit 


Longitude of 
the Node. 


Longitude of 
Perihelion. 


Perihelion 
Distance. 


Course. 


1456 
1531 
1607 
1682 


17° 56' 
17 56 
17 02 
17 42 


48° 30' 

49 25 

50 21 

50 48 


301° 00 

301 39 

302 16 

301 36 


0.58 
0.57 
0.58 
0.58 


Retrograde. 
Retrograde. 
Retrograde. 

Retrograde. 



262 COMETS. 

On comparing these elements, no doubt could be entertained 
that they belonged to one and the same body ; and since the 
interval between the successive returns was seen to be 75 or 76 
years, Halley ventured to predict that it would again return in 
1758. Accordingly, the astronomers who lived at that period 
looked for its return with 'the greatest interest. It was found, 
however, that on its way toward the sun it would pass very 
near to Jupiter and Saturn, and by their action on it, would 
be retarded for a long time. Clair aut, a distinguished French 
mathematician, undertook the laborious task of estimating the 
exact amount of this retardation, and found it to be no less 
than 618 days, namely, 100 days by the action of Jupiter, and 
518 days by that of Saturn. This would delay its appearance 
until early in the year 1759, and Clairaut fixed its arrival at 
the perihelion within a month of April 13th. It came to the 
perihelion on the 12th of March. 

401. The return of Halley's comet in 1835 was looked for 
with no less interest than in 1759. Several of the most accu- 
rate mathematicians of the age had calculated its elements 
with inconceivable labor. Their zeal was rewarded by the 
appearance of the expected visitant at the time and place 
assigned ; it traversed the northern sky, presenting the very 
appearances, in most respects, that had been anticipated ; and 
came to its perihelion on the 16th of November, within one 
day of the time predicted by Pontecoulant, a French mathe- 
matician, who had, it appeared, made the most successful cal- 
culation.* On its previous return, it was deemed an extraor- 
dinary achievement to have brought the prediction within a 
month of the actual time. 

Many circumstances conspired to render this return of Hal- 
ley's comet an astronomical event of transcendent interest. Of 
all the celestial bodies, its history was the most remarkable ; it 
afforded most triumphant evidence of the truth of the doctrine 
of universal gravitation, and consequently of the received laws 
of astronomy ; and it inspired new confidence in the power of 



* See Professor Loomis's Observations on Halley's Comet, Amer. Journ. Science, 
xxx., 209. Pontecoulant' s Phys. Celeste Precis, p. 586. 



ORBITS AND MOTIONS OF COMETS. 263 

that instrument (the Calculus) by means of which its elements 
had been investigated. 

402. Encke's comet, by its frequent returns, affords pecu- 
liar facilities for ascertaining the laws of its revolution ; and it 
has kept the appointments made for it with great exactness. 
On its return in 1839, it exhibited to the telescope a globular 
mass of nebulous matter resembling fog, and moved toward its 
perihelion with great rapidity. 

But what has made Encke's comet particularly famous, is its 
having first revealed to us the existence of a Resisting Medium 
in the planetary spaces. It has long been a question whether 
the earth and planets revolve in a perfect void, or whether a 
fluid of extreme rarity may not be diffused through space. A 
perfect vacuum was deemed most probable, because no such 
effects on the motions of the planets could be detected as indi- 
cated that they encountered a resisting medium. But a feather 
or a lock of cotton, propelled with great velocity, might render 
obvious the resistance of a medium which would not be per- 
ceptible in the motions of a cannon-ball. Accordingly, Encke's 
comet is thought to have plainly suffered a retardation from 
encountering a resisting medium in the planetary regions. 
The effect of this resistance, from the first discovery of the 
comet to the present time, has been to diminish the time of its 
revolution about two days. Such a resistance, by destroying 
part of tha projectile force, would cause the comet to approach 
nearer to the sun, and thus to have its periodic time shortened. 
The ultimate effect of this cause will be to bring the comet 
nearer to the sun at every revolution, until it finally falls into 
that luminary, although many thousand years will be required 
to produce this catastrophe.* It is conceivable, indeed, that 
the effects of such a resistance may be counteracted by the at- 
traction of one or more of the planets near which it may pass 
in its successive returns to the sun. It is peculiarly interesting 
to see a portion of matter of a tenuity exceeding the thinnest 
fog, pursuing its path in space, in obedience to the same laws 
as those which regulate such large and heavy bodies as Jupiter 

* Halley's comet, at its return in 1835, did not appear to be affected by the 
supposed resisting medium, and its existence is considered as still doubtful. 



264 COMETS. 

or Saturn. In a perfect void, a speck of fog, if propelled by a 
suitable projectile force, would revolve around the sun, and 
hold on its way through the widest orbit, with as sure and 
steady a pace as the heaviest and largest body in the system. 

403. The most remarkable comet of the present century 
hitherto observed, was the great comet of 1843. (See Plate I. 
at the end of the volume.) On the 28th of February of that 
year, the attention of numerous observers in various parts of 
the world was arrested by a comet seen in the broad light of 
day, a little eastward of the sun. In Mexico it was observed, 
and its altitude repeatedly measured with a sextant, from nine 
in the morning until sunset. In New England, it was seen at 
several places from half-past seven in the morning until after 
three in the afternoon, when the sky became obscured by hazi- 
ness and clouds. Accurate measures were taken by Capt. 
Clark, at Portland, Maine, of the distance of the nucleus from 
the sun's limb. At 3h. 2m. 15s. mean time, the distance of the 
sun's furthest limb from the nearest limb of the nucleus was 
4° 6' 15". The comet resembled a white cloud of great density, 
being nearly equally shining throughout, with a nucleus as 
bright as the full moon at midnight in a clear sky. During 
the first week in March, the appearance of this body was 
splendid and magnificent, enhanced in both respects by the 
transparency of a tropical sky, and the higher angle of eleva- 
tion above that at which it was seen by northern observers. 
At ISTew Haven, it was first seen after sunset, on the 5th. of 
March. It then lay far in the southwest. On account of the 
presence of the moon, it was not seen most favorably until the 
evening of the 17th. It then extended along the constellation 
Eridanus to the ears of the Hare, below the feet of Orion, reach- 
ing nearly to Sirius, being about 40° in length, although in the 
tropical regions its apparent length, at its maximum, was nearly 
70°. It was slightly curved like a goose-quill, and colored with 
a slight tinge of rose-red, which in a few evenings disappeared, 
and left it nearly a pearly white. Our diagram (Plate I. at the 
end of the volume) presents a pretty accurate idea of its ap- 
pearance on the 20th of March. All the astronomers of the 
age have agreed in the opinion that this is one of the most re- 
markable exhibitions of a comet ever witnessed, although they 



ORBITS AND MOTIONS OF COMETS. 265 

are not fully agreed respecting the elements of its orbit, or its 
periodic time. Its elements resemble those of the comet of 
1688, which would give a period of 175 years ; and to this pe- 
riodic time, authority at present inclines ;* but Prof. Hubbard, 
of the Washington Observatory, after an elaborate discussion 
of all the observations, thinks the most probable period 170 
years. 

Of all the comets on record, the great comet of 1843 ap- 
proached nearest to the sun. It came within about 60,000 
miles of his luminous surface, or only about one-fourth of the 
distance of the moon from the earth. It will be recollected 
that to a spectator on the earth the sun's angular diameter but 
a little exceeds half a degree ; but were we to approach as near 
to the sun as this body did in its perihelion, that diameter 
would appear no less than 121° 32' ; and the light and heat 
(which increase as the square of the distance is diminished) 
would be 47,000 times as great as at present, the heat exceed- 
ing nearly twenty-five times that produced by Parker's great 
burning lens, although this instrument is capable of producing 
effects beyond those of the most powerful blast-furnace. The 
velocity of the comet was still more astonishing, being at the 
rate of more than one and a quarter million of miles per hour, — 
a velocity sufficient to carry it through 180°, or half round the 
sun, in two hours. f 

An interesting comet appeared in 1858, called the comet of 
Donati, who first saw it at Florence, June 2d. , It continued in 
sight till Oct. 15th. Its tail, when at perihelion, Oct. 10th, 
was 60° long. . Its nucleus was uncommonly bright, and be- 
yond it, in the axis of the envelope and tail was a dark, straight 
line, like a shadow. The long period of its visibility gave un- 
usual opportunity for careful observations. Its periodic time is 
variously estimated, from 1,620 to 2,495 years.J 

404. Of the physical nature of comets, little is understood. 
It is usual to account for the variations which their tails un- 

• See American Almanac for 1844, p. 94. Amer. Journal of Science, xlv., 
p. 188. Astronomical Journal, vol. ii., p. 156. 

+ Herschel's Outlines, p. 318. 

| See Bond's "Account of Donati's Comet," with fine illustrations, in Math. 
Monthly, Dec, 1853. 



266 COMETS. 

dergo by referring them to the agencies of heat and cold. The 
intense heat to which they are subject in approaching so near 
the sun as some of them do, is alleged as a sufficient reason for 
the great expansion of the thin nebulous atmospheres forming 
their tails ; and the inconceivable cold to which they are sub- 
ject in receding to such a distance from the sun, is supposed to 
account for the condensation of the same matter until it returns 
to its original dimensions. The temperature experienced by 
the comets of 1680 and 1813 at their perihelion, would be suf- 
ficient to volatilize the most obdurate substances, and to ex- 
pand the vapor to vast dimensions ; and the opposite effects 
of the extreme cold to which it would be subject in the re- 
gions remote from the sun, would be adequate to condense it 
into its former volume. 

This explanation, however, does not account for the direction 
of the tail, extending, as it usually does, only in a line opposite 
to the sun. Some writers, therefore, as Delambre,* suppose 
that the nebulous matter of the comet, after being expanded 
to such a volume that the particles are no longer attracted to 
the nucleus unless by the slightest conceivable force, is carried 
off in a direction from the sun by the impulse of the solar rays 
themselves. Others conceive of a force of repulsion, independ- 
ent of any mechanical impulse emanating from the sun. But 
to assign such a power of communicating motion to the sun's 
rays while they have never been proved to have any momen- 
tum, or to a repulsive force which has no independent. proof 
of its existence, is unphilosophical ; and we are compelled to 
place the phenomena of comets' tails among the points of as- 
tronomy yet to be explained.-)* 

405. Since those comets which have their perihelion very 
near the sun, like the comet of 1680, cross the orbits of all the 
planets, the possibility that one of them may strike the earth, 
has frequently been suggested. Still, it may quiet our apprehen- 
sions on this subject, to reflect on the vast extent of the planet- 
ary spaces, in which these bodies are not crowded together as 



5 - 5 Delambre's Astronomy, t. iii., p. 401. 

f Prof. Norton on the "Formation of Comets' Tails," (Amer. Journal, 
xlvii., p. 104). 



METEORIC SHOWERS. 267 

we see tliein* erroneously represented in orreries and diagrams, 
but are sparsely scattered at immense distances from each 
other. They are like insects flying in the expanse of heaven. 
If a comet's tail lay with its axis in the plane of the eclip- 
tic when it was near the sun, we can imagine that the tail 
might sweep over the earth ; but the tail may be situated at 
any angle with the ecliptic as well as in the same plane with 
it, and the chances that it will not be in the same plane are 
almost infinite. It is also extremely improbable that a comet 
will cross the plane of the ecliptic precisely at the earth's path 
in that plane, since it may as probably cross it at any other 
point nearer or more remote from the sun. Still, some comets 
have occasionally approached near to the earth. Thus Biela's 
comet, in returning to the sun in 1832, crossed the ecliptic very 
near to the earth's track, and had the earth been then at that 
point of its orbit, it might have, passed through a portion of 
the nebulous atmosphere of the comet. The earth was within 
a month of reaching that point. This might at first view seem 
to involve some hazard ; yet we must consider that a month 
short implied a distance of nearly 50,000,000 miles. Laplace 
has assigned the consequences that would ensue in case of a di- 
rect collision between the earth and a comet ;* but terrible as 
he has represented them on the supposition that the nucleus of 
the comet is a solid body, yet considering a comet (as most of 
them doubtless are) as a mass of exceedingly light nebulous 
matter, it is not probable, even were the earth to make its way 
directly through a comet, that a particle of the comet would 
reach the earth. * The portions encountered by the earth, would 
be arrested by the atmosphere, and probably inflamed ; and 
they would perhaps exhibit, on a more magnificent scale than 
was ever before observed, the phenomena of shooting stars or 
meteoric showers. 

METEORIC SHOWERS. 

406. The remarkable exhibitions of shooting stars which 
have occurred within a few years past, have excited great in- 
terest among astronomers, and led to some new views respect- 
ing the construction of the solar system. Their attention was 

8 Syst. du Monde, 1. iv. T c. 4. 



268 METEORIC SHOWERS. 

first turned toward this subject by the great meteoric shower 
of November 13, 1833, On that morning, from two o'clock 
until broad daylight, the sky being perfectly serene and cloud- 
less, the whole heavens were lighted with a magnificent dis- 
play of celestial fireworks. Numerous bright bodies, which 
might be compared with stars of the largest magnitudes, and 
with planets, were darting toward the earth on all sides, describ- 
ing arcs of great circles, of all lengths from 70° to less than a 
single degree. In many cases, they left long trains of light in 
their paths, which lasted a few seconds ; and occasionally, when 
a meteor of unusual brightness descended, the train of light con- 
tinued for minutes. The light which some of them shed was 
equal to that of the moon at the quarter. The whole number 
seen at any one place of observation could not have been less 
than 200,000. 

On tracing back the lines of direction in which the meteors 
moved, it was found that they all appeared to radiate from the 
same point, which was situated near one of the stars {Gamma 
Leonis) of the sickle, in the constellation Leo; and, in every 
repetition of the meteoric shower of November, the radiant 
point has occupied nearly the same situation. 

This shower pervaded nearly the whole of North America, 
having appeared in almost equal splendor from the British 
possessions on the north, to the West India Islands and Mexico 
on the south, and from sixty-one degrees of longitude east of 
the American coast, quite to the Pacific ocean on the west. 
Throughout this immense region the duration was nearly the 
same. The meteors began to attract attention by their un- 
usual frequency and brilliancy, from nine to twelve o'clock in 
the evening ; were most striking in their appearance from two 
to four y arrived at their maximum, in many places, about 
four o'clock ; and continued until rendered invisible by the 
light of day. The meteors moved in right lines, or in such 
apparent curves, as, upon optical principles, can be resolved 
into right lines. Their general tendency was toward the north- 
west, although by the effect of perspective they appeared to 
move in all directions. 

407. Soon after this occurrence, it was ascertained that a 
similar meteoric shower had appeared in 1799, and what was 



METEORIC SHOWERS. 269 

remarkable, almost exactly at the same time of the year, name- 
ly, on the morning of the 12th of November; and it soon ap- 
peared, by accounts received from different parts of the world, 
that this phenomenon had occurred on the same 13th of No- 
vember, in 1830, 1831, and 1832. Hence, this was evidently 
an event independent of the casual changes of the atmosphere : 
for, having a periodical return, it was undoubtedly to be re- 
ferred to astronomical causes, and its recurrence, at a certain 
definite period of the year, plainly indicated some relation to 
the revolution of the earth around the sun. 

It remained, however, to develop the nature of this relation, 
by investigating, if possible, the origin of the meteors. The 
views to which the author of this work was led, suggested the 
probability that the same phenomenon would recur at the 
corresponding seasons of the year for at least several years 
afterward ; and such proved to be the fact, although the 
appearances, at every succeeding return, were less and less 
striking, until 1839, when, so far as is known, they ceased 
altogether. 

Meanwhile, three other distinct periods of meteoric showers 
have also been determined ; one on the 9th of August, and 
(more rare) on the 21st of April and 7th of December respect- 
ively. 

408. The following conclusions respecting the meteoric 
shower of November, are believed to be well established, and 
most of them are now generally admitted by astronomers, 
though we can not here exhibit the evidence on which they 
were founded.* 

It is considered, then, as established, that the periodical 
meteors of November (and most of the conclusions apply 
equally to those of August) have their origin beyond the 
atmosphere, descending to us from some body (which, from 
the known constitution of the meteors, may be called a nebu- 
lous body) with which the earth falls in, and near or through 
the borders of which it passes ; that this body has an inde- 
pendent existence as a member of the solar system, its periodic 

° We beg leave to refer the reader to various publications on the subject, by 
the author and others, in the American Journal of Science, commencing with the 
25th volume ; and also to " Letters on Astronomy," by the author of this work. 



270 METEORIC SHOWERS. 

time being nearly commensurable with the earth's, either a 
year or half a year, so that for a number of years in succession 
the two bodies meet near the same part of the earth's orbit. 
Tt is further established, that the meteors consist of light com- 
bustible matter; that they move with great velocities, amount- 
ing, in some instances, to not less than that of the earth in 
its orbit, or 19 miles per second ; that some of them are bodies 
of large size, sometimes several thousand feet in diameter ; 
that when they enter the atmosphere, they rapidly and power- 
fully condense the air before them, and thus elicit the heat 
that sets them on fire, as a spark is elicited in the air-match, 
by being suddenly condensed by means of a piston and cylin- 
der ; and that they are burned up at a considerable height 
above the earth, sometimes not less than 30 miles. 

409. Calling the body from which the meteors descended 
the "meteoric body," it is inferred that it is a body of great 
extent, since, without apparent exhaustion, it has been able to 
afford such copious showers of meteors at so many different 
times ; and hence we regard the part that has descended to the 
earth only as the extreme portions of a body or collection of 
meteors, of unknown extent, existing in the planetary spaces. 
Since the earth fell in with the meteoric body, in the same 
part of its orbit for several years in succession, the body must 
either have remained there while the earth was performing its 
whole revolution around the sun, or it must itself have had a 
revolution, as well as the earth. No body can remain station- 
ary within the planetary spaces ; for, unless attracted to some 
nearer body, it would be drawn directly toward the sun, and 
could not have been encountered by the earth again in the 
same part of her orbit. • E"or can any mode be conceived in 
which this event could have happened so many times in regu- 
lar succession, unless the body had a revolution of its own 
around the sun. Finally, to have come into contact with the 
earth at the same part of her orbit, in two or more successive 
years, the body must have a period which is either nearly the 
same with the earth's period, or some aliquot part of it. ~No 
period will fulfil these conditions, but either a year or half a 
year. Which of these is the true period of the meteoric body, 
is not fully determined. 



PART III. -OF THE FIXED STARS AND SYSTEM OF THE 

WORLD. 



CHAPTER I. 

OF THE FIXED STARS — CONSTELLATIONS. 

410. The Fixed Stars are so called, because, to common 
observation, they always maintain the same situations with re- 
spect to one another. 

The stars are classed by their apparent magnitudes. The 
whole number of magnitudes recorded are sixteen, of which the 
first six only are visible to the naked eye ; the rest are telescopic 
stars. As the stars which are now grouped together under one 
of the first six magnitudes are very unequal among themselves, 
it has recently been proposed to subdivide each class into three, 
making in all eighteen instead of six magnitudes visible to the 
naked eye. These magnitudes are not determined by any 
very definite scale, but are merely ranked according to their 
relative degrees of brightness, and this is left in a great meas- 
ure to the decision of the eye alone, although it would appear 
easy to measure the comparative degree of light in a star by a 
photometer, and upon such measurement to ground a more 
scientific classification of the stars. The brightest stars to the 
number of 15 or 20 are considered as stars of the first magni- 
tude ; the 50 or 60 next brightest, of the second magnitude ; 
the next 200 of the third magnitude ; and thus the number of 
each class increases rapidly as we descend the scale, so that no 
less than fifteen or twenty thousand are included within the 
first seven magnitudes. 

411. The stars have been grouped in Constellations from 
the most remote antiquity ; a few, as Orion, Bootes, and Ursa 
Major, are mentioned in the most ancient writings under the 
same names as they bear at present. The names of the con- 



272 FIXED STARS. 

stellations are sometimes founded on a supposed resemblance 
to the objects to which the names belong; as the Swan and 
the Scorpion were evidently so denominated from their like- 
ness to those animals ; but in most cases it is impossible for us 
to find any reason for designating a constellation by the figure 
of the animal or the hero which is employed to represent it. 
These representations were probably once blended with the 
fables of pagan mythology. The same figures, absurd as they 
appear, are still retained for the convenience of reference ; 
since it is easy to find any particular star, by specifying the 
part of the figure to which it belongs, as when we say a star is 
in the neck of Taurus, in the knee of Hercules, or in the tail of 
the Great Bear. This method furnishes a general clue to its 
position ; but the stars belonging to any constellation are dis- 
tinguished according to their apparent magnitudes, as follows : 
first, by the Greek letters, Alpha, Beta, Gamma, &e. Thus a 
Ononis, denotes the largest star in Orion, (3 Andromeda?, the 
second star in Andromeda, and y Zeonis, the third brightest 
star in the Lion. Where the number of the Greek letters is 
insufficient to include all the stars in a constellation, recourse 
is had to the letters of the Roman alphabet, a, b, c, &c. ; and 
in cases where these are exhausted, the final resort is to num- 
bers. This is evidently necessary, since the largest constella- 
tions contain many hundreds or even thousands of stars. Cat- 
alogues of particular stars have also been published by different 
astronomers, each author numbering the individual stars em- 
braced in his list, according to the places they respectively 
occujyy in the catalogue. These references to particular cata- 
logues are sometimes entered on large celestial globes. Thus 
we meet with a star marked 84 H., meaning that this is its 
number in Herschel's catalogue, or 140 M., denoting the place 
the star occupies in the catalogue of Mayer. 

412. The earliest catalogue of the stars was made by Hip- 
parchus, of the Alexandrian school, about 140 years before the 
Christian era. A new star appearing in the firmament, he was 
induced to count the stars and to record their positions, in order 
that posterity might be able to judge of the permanency of the 
constellations. His catalogue contains all that were conspicu- 
ous to the naked eve in the latitude of Alexandria, being 1,022. 



CONSTELLATIONS. 273 

Most persons unacquainted with the actual number of the stars 
which compose the visible firmament, would suppose it to be 
much greater than this ; but it is found that the catalogue of 
Hipparchus embraces nearly all that are easily seen in the same 
latitude, and that on the equator, where the spectator has the 
northern and southern hemispheres both in view, the number 
of stars that can be counted does not exceed 3000. A hasty 
glance over the sky gives us the impression of a countless mul- 
titude of stars ; but the greater part vanish as soon as we try 
to number them. This is owing to the indirect vision of 
thousands of faint stars, which are unseen as soon as we turn 
the axes of the eyes directly upon them. 

By the aid of the telescope, new fields of stars present them- 
selves of boundless extent ; the number continually augmenting 
as the powers of the telescope are increased. Lalande, in his 
Histoire Celeste, has registered the positions of no less than 
50,000 ; and the whole number visible in the largest telescopes 
amount to many millions. 

413. It is strongly recommended to the learner to acquaint 
himself with the leading constellations at least, and with a few 
of the most remarkable individual stars. The task of learning 
them is comparatively easy, when they are taken up at suitable 
intervals throughout the year, the moon being absent and the 
sky clear. After becoming familiar with such constellations 
as are visible on any given evening (suppose the first of Jan- 
uary), these may be carefully reviewed after an interval of a 
month, and the several new ones added which have in the 
mean time risen above the eastern horizon. By repeating this 
process near the beginning of every month of the year, the 
learner will acquire a competent knowledge of the whole that 
are visible in his latitude, and with a small expenditure of time. 
It may at first be advisable to obtain, for an evening or two. 
the assistance of some one who is acquainted with the constel- 
lations, to point out such as are then visible in the evening sky. 
Then, by the aid of a celestial map, or, what is better, a celes- 
tial globe, the learner will pursue the study without difficulty. 
We begin by rectifying the globe for the time, according to the 
directions given in Article 76. 

In the following sketch of the leading constellations, we will 

is 



274 FIXED STARS. 

point out a few of the marks by which they may be severally 
recognized, adding occasionally a few particulars, and leaving 
it to the learner to fill up the outline by the aid of his map Oi 
globe, one of which, indeed, is presumed to be before him.* 

Let us begin with the constellations of the Zodiac, which, 
succeeding each other as they do in a known order, are most 
easily found.f 

Aries (the Ram), the first constellation of the Zodiac, is 
known by two bright stars, Alpha (a) on the northeast, and 
Beta (|3) on the southwest, 4°^: apart, forming the head. South 
of Beta, at the distance of 2°, is a smaller star, Gamma (y). 
The next brightest star of the Ram, Delta ((5), is in the tail, 
15° southeast of Alpha. The feet of the figure rest on the 
head of the Whale. It has been already intimated (Art. .193), 
that the vernal equinox was near the head of Aries, when the 
signs of the Zodiac received their present names, but that the 
equinox is now found 30° westward of a Arietis, in consequence 
of the precession of the equinoxes. 

Taurus (the Bull) will be readily found by the seven stars, 
or Pleiades, which lie in the neck, 24° eastward of a Arietis. 
The largest star in Taurus is Aldebaran, of the first magnitude, 
in the Bull's eye, 14° southeast of the Pleiades. It has a red- 
dish color, and resembles the planet Mars. The other eye of 



* A celestial globe, sufficient for studying the constellations, may be purchased 
for a small sum, and is, in other respects, a valuable possession to the astronom- 
ical student ; but even cheap maps of the stars, like those of Burritt or Kendal, 
will answer for beginners ; and the Celestial Atlas, published by the Society for 
the Diffusion of Useful Knowledge, which is suitable for the more advanced 
student, may be procured at a moderate expense. 

f It will be expedient, where it is practicable, for the learner to study the con- 
stellations in separate portions, at different seasons of the year, as at the equi- 
noxes and at the solstices, according to the directions given in the closing article 
of this chapter. 

\ These measures are not intended to be stated with minute accuracy, but only 
with such a degree of exactness as may serve for a general guide. The learner 
will find it greatly for his advantage to accustom himself to make an accurate es- 
timate with the eye of distances in degrees on the celestial sphere ; and he may, 
at the outset, fix on the distance between Alpha and Beta Arietis as a standard 
measure (4°) by which to estimate other angular distances among the stars. 
Thus, half this length applied from Beta to Gamma, indicates that the two latter 
stars are 2° apart ; and two and a half times the same measure (10°) will reach 
from the Pleiades to Aldebaran. Or the Pointers in the Great Bear will furnish 
a measure of 5°. 



CONSTELLATIONS. 275 

the figure is EpsRon (e), 3° northwest of Aldebaran. Five small 
stars, situated a little west of Aldebaran, in the face of the 
Bull, constitute the Hyade-s. Although the Pleiades are usually 
denominated the seven stars, jet it has been remarked, from a 
high antiquity, that only six are present. 

Quse septem dici, sex tamen esse solent.* — Ovid. 

Some persons, however, of remarkable powers of vision, are 
still able to recognize seven, and even a greater number, f 
With a moderate telescope, not less than 50 or 60 stars, of con- 
siderable brightness, may be counted in this group, and a much 
larger number of very small stars are revealed to the more 
powerful telescopes. The beautiful allusion, in the book of 
Job, to the " sweet influences of the Pleiades," and the special 
mention made of this group by Homer and Hesiod, show how 
early it had attracted the attention of mankind. The horns of 
the Bull are two stars, Beta and Zeta, situated 25° east of the 
Pleiades, being 8° apart. The northern horn, Beta, also forms 
one of the feet of Auriga, the Charioteer. 

Gemini (the Twins) is represented by two well-known stars, 
Castor and Pollux, in the head of the figure, 5° asunder. 
Castor, the northern, is of the first, and Pollux of the second 
magnitude. Four conspicuous stars, extending in a line from 
south to north, 25° S. W. of Castor, form the feet, and two 
others, parallel to these at the distance of six or seven degrees 
northeastward, are in the knees. 

Cancer (the Crab). There are no large stars in this constel- 
lation, and it is regarded as less remarkable than any other in 
the Zodiac. The two most conspicuous stars, Alpha and Beta, 
are in the southern claws of the figure, and in its body are the 
northern and southern Asellus, which may be readily found 
on a celestial globe. But the most remarkable object in this 
constellation is a misty group of very small stars, so close to- 
gether, when seen by the naked eye, as to resemble a comet, 



* Their names were Electra, Maia, Taygeta, Alcyone, Celreno. Asterope, and 
Merope, the last being the "Lost Pleiad" of the poets. Alcyone, according to 
a recent celebrated hypothesis, is distinguished as the center around which the 
starry host revolve. 

f Smyth's Cycle, ii., p. 86. 



276 FIXED STARS. 

but easily separated by the telescope into a beautiful collection 
of brilliant points. It is called Prcesepe, or the Beehive. 

Leo (the Lion) is a very large constellation, and has many 
interesting members. Regulus (a Leonis) is a star of the first 
magnitude, which lies very near the ecliptic, and is much used 
in astronomical observations. North of Regulus lies a semi- 
circle of five bright stars, arranged in the form of a sickle, of 
which Regulus is the handle, and extending over the shoulder 
and neck of the Lion.* Denebola, a conspicuous star in the 
Lion's tail, lies 25° east of Regulus. Twenty bright stars in all 
help to compose this beautiful constellation. It ranges from 
west to east along the Zodiac, over more than 40° of longi- 
tude, all parts of the figure excepting the feet lying north of 
the ecliptic. 

Yirgo (the Virgin) extends along the Zodiac eastward from 
the Lion, covering an equally wide region of the heavens, al- 
though less distinguished by brilliant stars. Spica, however, 
is a star of the first magnitude, and lies a little east of the ver- 
nal equinox. Vindemiatrix, in the arm of Yirgo, 18° east of 
Denebola, and 23° north of Spica, is easily found, and directly 
south of Denebola 13°, is (3 Virginis ; while four other conspic- 
uous stars, in the form of a trapezium, between this and Yin- 
demiatrix, lie in the wing and shoulders of the figure. The 
feet are near the Balance. 

Libra (the Balance) is composed of a few scattered mem- 
bers situated between the feet of Yirgo and the head of Scor- 
pio, but has no very distinctive marks. Two stars of the 
second magnitude, Alpha on the south, and Beta 8° northeast 
of Alpha, together with a few smaller stars, form the scales. 

Scorpio (the Scorpion) is one of the finest of the constella- 
tions of the Zodiac, and is manifestly so called from its resem- 
blance to the animal whose name it bears. The head is com- 
posed of five stars, arranged in a line slightly curved, which is 
crossed in the center by the ecliptic, nearly at right angles, a 
degree south of the brightest of the group (3 Scorpionis. Nine 
degrees southeast of this is a remarkable star of the first mag- 



* As the Meteors of November always appear to radiate from a point in the bend 
of the sickle, near the star Gamma, it may be noted that the names of the six 
stars composing this figure, beginning with Reguius, are a, n, y, £, n, e. 



CONSTELLATIONS. 277 

nitude, called Antares, and sometimes the Heart of the Scor- 
pion (Cor Scovpioms). It is of a red color, resembling the 
planet Mars. South and east of this, a succession of not less 
than nine bright stars sweep round in a semicircle, terminating 
in several small stars forming the sting of the Scorpion. The 
tail of the figure extends into the Milky Way. 

Sagittarius (the Aechee). Ten degrees eastward of the 
Scorpion's tail, on the eastern margin of the Milky Way, we 
come to the bow of Sagittarius, consisting of three stars about 
6° apart, the middle one being the brightest, and situated in 
the bend of the bow, while a fourth star, 4° westward of it, 
constitutes the arrow. The archer is represented by the figure 
of a Centaur (half horse and half man), and proceeding about 
ten degrees east from the bow, we come to a collection of seven 
or eight stars of the second and third magnitudes, which lie in 
the human or upper part of the figure. 

Capeicoenus (the Goat), represented with the head of a goat 
and the tail of a fish, comes next to Sagittarius, about 20° east- 
ward of the group that form the upper portions of that con- 
stellation. Two stars of the second magnitude, a on the north, 
and /3 on the south, 3° apart, constitute the head of Caprieor- 
nus, while a collection of stars of the third magnitude, lying 
20° southeast of these form the tail. 

Aquaeics (the Watee Beasee) is closely in contact with 
the tail of Capricornus, immediately north of which, at the 
distance of 10°, is the western shoulder (/I), and 10° further 
east is the eastern shoulder (a) of Aquarius. About 3° south- 
east of a is y Aquarii, which, together with the other two, 
makes an acute triangle, of which fi forms the vertex. In the 
eastern arm of Aquarius are found four stars, which together 
make the figure Y, the open part being westward, or toward 
the shoulders of the constellation. Aquarius ranges nearly 
30° from north to south, being nearly bisected by the ecliptic. 

Pisces (the Fishes). Three figures of this kind, at a great 
distance apart, two north and one south of the ecliptic, com- 
pose this constellation. The Southern Fish, Piseis Austral is, 
otherwise called Fomalhaut, lies directly below the feet of 
Aquarius, and being the only conspicuous star in that part of 
the heavens, is much used in astronomical measurements. It 
is 30° south of the equator. 



278 FIXED STAES. 

About 12° east of the figure Y in the arm of Aquarius, is an 
assemblage of five stars, forming a pretty regular pentagon, 
which is one of the northern members of the constellation 
Pisces ; and far to the northeast of this figure, north of the 
head of Aries, lies the third member, the three being repre- 
sented as connected together by a -ribbon, or wavy band, com- 
posed of minute stars. 

414. The Constellations of the Zodiac being first well 
learned, so as to be readily recognized, will facilitate the learn- 
ing of others that lie north and south of them. Let us there- 
fore next review the principal Northern Constellations, begin- 
ning at the North Pole. 

Ursa Minor (the Little Bear). The Pole-star {Polaris) 
is in the extremity of the tail of the Little Bear. It is of the 
third magnitude, and being within less than a degree and a 
half of the North Pole of the heavens, it serves at present to 
indicate the position of the pole. It will be recollected, how- 
ever, that on account of the precession of the equinoxes, the 
pole of the heavens is constantly shifting its place from east to 
west, revolving about the pole of the ecliptic, and will in time 
recede so far from the pole-star, that this will no longer retain 
its present distinction (Art. 190). Three stars in a straight 
line, 4° or 5° apart, commencing with Polaris, lead to a trape- 
zium of four stars, the whole seven together forming the figure 
of a dipper, the trapezium being the body, and the three first- 
mentioned stars being the handle. 

Ursa Major (the Great Bear) is one of. the largest and 
most celebrated of the constellations. It is usually recognized 
by the figure of a larger and more perfect dipper than the one 
in the Little Bear — three stars, as before, constituting the han- 
dle, and four others, in the form of a trapezium, the body of 
the figure. The two western stars of the trapezium, ranging 
nearly with the North Star, are called the Pointers • and be- 
ginning with the northern of these two, and following round 
from left to right through the whole seven, they correspond in 
rank to the succession of the first seven letters of the Greek 
alphabet, Alpha. Beta, Gamma, Delta, Epsilon, Zeta, Eta. 
Several of them also are known by their Arabic names. Thus, 
the first in the tail, corresponding to Epsilon, is Alioth, the 



CONSTELLATIONS. 279 

next (Zeta) Mizar, and the last (Eta) Benetnasch. These are 
all bright and beautiful stars, Alpha being of the first magni- 
tude; Beta, Gamma, Delta, of the second; and the three form- 
ing the tail, of the third. But it must be remarked that this 
very remarkable figure of a dipper or ladle composes but a 
small part of the entire constellation, being merely the hinder 
half of the body and the tail of the Bear. The head and breast 
of the figure, lying about ten or twelve degrees west of the 
Pointers, contain a great number of minute stars in a triangu- 
lar group. One of the fourth magnitude, Omicron, is in the 
mouth of the Bear. The feet of the figure may be looked for 
about 15° south of those already described, the two hinder 
paws consisting each of two stars very similar in appearance, 
and only a degree and a half apart. The two paws are distant 
from each other about 18° ; and following westward about the 
same number of degrees, we come to another very similar pair 
of stars, which constitute one of the fore paws, the other foot 
being without any corresponding pair. 

In a clear winter's night, when the whole constellation is 
above the pole, these various parts may be easily recognized, 
and the entire figure will be seen to resemble a large animal, 
readily accounting for the name given to this constellation 
from the earliest ages. 

Deaco {the Deagon) is also a very large constellation, ex- 
tending for a great length from east to west. Beginning at 
the tail which lies half way between the Pointers and the Pole- 
star, and winding round between the Great and the Little 
Bear, by a continued succession of bright stars from 5° to 10° 
asunder, it coils around under the feet of the Little Bear, 
sweeps round the pole of the ecliptic, and terminates in a 
trapezium formed by four conspicuous stars, from thirty to 
thirty-five degrees from the North Pole. A few of the mem- 
bers of this constellation are of the second, but the greater part 
of the third magnitude, and below it. 

415. "With the constellations already described as general 
landmarks, we may now proceed with each of the principal 
remaining ones, by stating its boundaries, as we do those of 
countries in geography ; their relative situations being thus 
first learned from a ma}), or (what is better) from a celestial 



280 FIXED STARS. 

globe, and then being severally traced out on the sky itself. 
"We will begin with those which surround the North Pole. 

Cepheus (the King) is bounded N. by the Little Bear, E. by 
Cassiopeia, S. by the Lizard, and W. by the Dragon. The 
head lies in the Milky Way, and the feet extend toward the 
pole. It contains no stars above the third magnitude. 

Cassiopeia is bounded N. and W. by Cepheus, E. by Camel- 
opardalus, and S. by Andromeda, and is one of the constella- 
tions of the Milky Way. It is readily distinguished by the 
figure of a chair inverted, of which two stars constitute the 
back, and four, in the form of a square, the body of the chair. 
It is on the opposite side of the pole from the Great Bear, and 
nearly at the same distance from it. 

Camelopardalus (the Giraffe) is bounded N. by the Little 
Bear, E. by the head of the Great Bear, S. by Auriga and Per- 
seus, and W. by Cassiopeia. Although this constellation oc- 
cupies a large space, yet it has no conspicuous stars. 

Andkomeda is bounded N. by Cassiopeia, E. by Perseus, S. 
by Pegasus, and W. by the Lizard. The direction of the figure 
is from S. W. to N. E., the head coming down within 30° of 
the equator, and being recognized by a star of the second mag- 
nitude, which forms the northeastern corner of the great square 
in Pegasus, to be described hereafter. At the distance of six 
or seven degrees from the head, are three conspicuous stars in 
a row, ranging from north to south, which lie in the breast of 
the figure ; and about the same distance from these, and par- 
allel to them, three more, which constitute the girdle of An- 
dromeda. Near the northernmost of the three, is a faint, 
misty object, often mistaken for a comet, but is a nebula, and 
one of the most remarkable in the heavens. 

Pekseus is bounded 1ST. by Cassiopeia, E. by Auriga, S. by 
Taurus, and W. by Andromeda. The figure extends from 
north to south, and is represented by a giant holding aloft a 
sword in his right hand, while his left grasps the head of Me- 
dusa, — a group of stars on the western side of the figure, em- 
bracing the celebrated star Algol. A series of bright stars de- 
scend along the shoulders and the waist, and there divide into 
the two legs. The western foot is 8° degrees north of the 
Pleiades. The eastern leg is bent at the knee, which is distin- 
guished by a group of small stars. Near the sword handle, 



CONSTELLATIONS. 281 

under Cassiopeia's chair, is a fine cluster of stars, so close to- 
gether as scarce to be separable by the eye. 

Auriga (the Wagoner) is bounded 1ST. by Camelopardalus, 
E. by the Lynx, S. by Taurus, and W. by Perseus. He is rep- 
resented as bearing on his left shoulder the little Goat Cajpella, 
a white and beautiful star of the first magnitude, (a Aurigse), 
while Beta forms the right shoulder, S° east of Capella. These 
two bright stars form, with the northern horn of the Bull, at 
the distance of 18°, an isosceles triangle. 

Leo Minor (the Lesser Lton) is bounded !N". by Ursa Ma- 
jor, E. by Coma Berenices, S. by Leo, and W. by the Lynx. 
It lies directly under the hind feet of the Great Bear, and over 
the sickle in Leo, and is easily distinguished. Four stars in 
the central part of the figure, from 4° to 5° apart, form a pretty 
regular parallelogram. 

Canes Yenatici (the Greyhounds). This constellation lies 
between the hind legs of the Great Bear on the west, and 
Bootes on the east ; Cor Caroli, a solitary star of the third 
magnitude, 18° south of Alioth, in the tail of the Great Bear, 
will serve to mark this constellation. 

Coma Berenices (Berenice's Hair) is a cluster of small 
stars, composing a rich group, 15° N. E. of Denebola, in the 
Lion's tail, in a line between this star and Cor Caroli, and half 
way between the two. 

Bootes is bounded N. by Draco, E. by the Crown and the 
head of Serpentarius, S. by Virgo, and W. by Coma Berenices 
and the Hounds. It reaches for a great distance from north to 
south, the head being within 20° of the Dragon, and the feet 
reaching to the Zodiac. In the knee of Bootes is Arcturus, a 
star of the first magnitude. The next brightest star, Beta, is 
in the head of Bootes, 23° north of Arcturus, and 15° east of 
the last star in the tail of the Great Bear. 

Corona Borealis (the Northern Crown) is bounded "N. 
and E. by Hercules, S. by the head of Serpentarius, and W. by 
Bootes. It is formed of a semicircle of bright stars, six in 
number, of which Gemma, near the center of the curve, is of 
the second magnitude. 

Hercules is bounded N". by Draco, E. by Lyra, S. by Ophi- 
uchus, and W. by Corona Borealis. It is a very large constel- 
lation, and contains some brilliant objects for the telescope, al- 



282 FIXED STARS. 

though its components are generally very small. The figure 
lies north and south, with the head near the head of Ophiu- 
chus, and the feet under the head of Draco. Being between 
the Crown and the Lyre, its locality is easily determined. The 
eastern foot of Hercules forms an isosceles triangle with the two 
southern stars of the trapezium in the head of Draco ; while 
the head of Hercules is far in the south, within 15° of the 
equator, being 6° west of a similar star which constitutes the 
head of Ophiuchus. 

Lyra (the Lyre) is bounded U. by the head of Draco, E. by 
the Swan, S. and W. by Hercules. Alpha Lyrse, or Yega y is 
of the first magnitude. It is accompanied by a small acute 
triangle of stars. Its color is a shining white, resembling Ca- 
pella and the Eagle. 

Cyg-nus (the Swan) extends along the Milky Way, below 
Cepheus, and immediately eastward of the Lyre, and has the 
figure of a large bird flying along the Milky Way from north 
to south, with outstretched wings and long neck. Commen- 
cing with the tail 25° east of Lyra, and following down the 
Milky Way, we pass along a line of conspicuous stars which 
form the body and neck of the figure ; and- then returning to 
the second of the series, we see two bright stars at eight or 
nine degrees on the right and left (the three together ranging 
across the Milky Way) which form the wings of the Swan. 
This constellation is among the few which exhibit some resem- 
blance to the animals whose names they bear. 

Vulpecula (the Little Fox) is a small constellation, in 
which a fox is represented as holding a goose in his mouth. It 
lies in the Milky Way, between the Swan on the north and 
the Dolphin and the Arrow on the south. 

Aquila (the Eagle) stretches across the Milky Way, and is 
bounded 1ST. by Sagitta, a small constellation which separates 
it from the Fox, E. by the Dolphin, S. by Antinous, and W. by 
Taurus Poniatowski (the Polish Bull), which separates it from 
Ophiuchus. It is distinguished by three bright stars in the neck, 
known as the " three stars," which lie in a straight line about 2° 
apart, on the eastern margin of the Milky Way. The central 
star is of the first magnitude. Its Arabic name is Altair. 

Antinous lies across the equator, between the Eagle on the 
north and the head of Capricorn on the south. 



CONSTELLATIONS. 283 

Delpbtnus (the Dolphin) is situated east and north of Al- 
tair, and is composed of five stars of the third magnitude, of 
which four in the form of a rhombus, compose the head, and 
the fifth forms the tail. 

Pegasus (the Flying Horse) is a very large constellation, 
and is bounded !N~. by the Lizard and Andromeda, E. and S. 
by Pisces, W. by the Dolphin. The head is near the Dolphin, 
while the back rests on Pisces, and the feet extend toward 
Andromeda. 

A large square, composed of four conspicuous members, one 
(3farJcab) of the first, and three others of the second magnitude, 
distinguish this constellation. The corners of the square are 
about 15° apart ; the northeastern corner being in the head of 
Andromeda. 

Ophiuchus is another very large constellation, the head 
being near the head of Hercules, and the feet reaching to Scor- 
pio, the western foot being almost in contact with Antares. 
The figure is that of a giant holding* a serpent in his hands. 
The head of the serpent is a little south of the Crown, and the 
tail reaches far eastward toward the Eagle. 

416. Of the Constellations which lie south of the Zodiac, 
we shall notice only Cetus, Orion, Lepus, Monoceros, Canis 
Major, Canis Minor, Hydra, Crater, and Corvus. 

Cetus (the Whale) is distinguished rather for its extent than 
its brilliancy t occupying a large tract of the sky south of the 
constellations Pisces and Aries. The head is directly below 
the head of Aries, and the tail reaches westward 45°, being 
about 10° south of the vernal equinox. Menkar (« Ceti), the 
largest of its components, is situated in the mouth, 25° south- 
east of a Arietis; and Mira (o Ceti) in the neck, 14° west of 
Menkar, is celebrated as a variable star, which exhibits differ- 
ent magnitudes at different times. 

Orion is one of the most magnificent of the constellations, 
and one of those that have longest attracted the admiration of 
mankind, being alluded to in the book of Job, and mentioned 
by Homer. The head of Orion lies southeast of Taurus, 15° 
from Aldebaran, and is composed of a cluster of small stars. 
Two very bright stars, Betalgeuse of the first, and BeUatrix of 
the second magnitude, form the shoulders ; three more, re- 



284: FIXED STARS. 

sembling the three stars of the eagle, compose the girdle ; and 
three smaller stars, in a line inclined to the girdle, form the 
sword. Rigel) of the first magnitude, makes the west foot, bnt 
the corresponding star, 9° southeast of this, which is sometimes 
taken for the other foot, is above the knee, this foot being con- 
cealed behind the Hare. Orion's club is marked by three stars 
of the fifth magnitude, close together, in the Milky Way, just 
below the southern horn of the Bull. Orion is a favorite con- 
stellation with the practical astronomer, abounding, as it does, 
in addition to the splendor of its components, with fine nebulae, 
double stars, and other objects of peculiar interest when viewed 
with the telescope. It embraces 70 stars, plainly visible to the 
naked eve, including two of the first, four of the second, and 
three of the third magnitude. 

Li-pus (the Hake). Below Pigel, the western foot of Orion, 
is a small trapezium of stars, which forms the ears of the Hare; 
and an assemblage of nine stars, of the third and fourth mag- 
nitudes, south and east of these, make up the remaining parts 
of the figure. 

Canis Major (the Greater Dog) lies directly east of the 
Hare, and is highly distinguished by containing Sirius, the 
most splendid of all the fixed stars, which lies in the mouth of 
the figure. In the fore paw, 6° west of Sirins, is a star of the 
second magnitude (/3 Can is Majoris), and from 10° to 15° south 
of Sinus, is a collection of stars of the second and third magni- 
tudes, which make up the hinder portions of the figure. The 
Egyptians, who anticipated the rising of the Nile by the ap- 
pearance of Sirius in the morning sky, represented the constella- 
tion by the figure of a dog, the symbol of a faithful watchman. 

Canis Minor (the Lesser Dog). About 25° north of Sirius, 
is the bright star Procyon, also of the first magnitude, which 
marks the side of the Lesser Dog. A star of the third magni- 
tude (3), 4° northwest of this, in the head of the figure, forms 
with Procyon the lower side of an elongated parallelogram, of 
which Castor and Pollux, 25° north, form the upper side. 

Monoceros is a large constellation, occupying the space be- 
tween the Greater and the Lesser Dog, but has no conspicuous 
members. 

Hydra occupies a long space south of Leo, Virgo, and Libra. 
Its head, which is south of the fore paws of the Lion, consists 



CONSTELLATIONS. 285 

of four stars of the fourth magnitude, of nearly uniform appear- 
ance ; and about 15° S. E. of these is the Heart {Cor Hydrce), 
23° south of Regulus. Resting on Hydra, and south of the 
hind feet of Leo, is Crater (the Cup), consisting of six stars of 
the fourth magnitude, arranged in the form of a semicircle; 
and a little further east, also perched on the back of Hydra, is 
Corvus (the Croio), the two brightest components of which are 
situated in one of the wings of the figure, in a line between 
Crater and Spica Yirginis. 

417. According to an intimation given in a note on p. 274, 
the constellations may be advantageously studied at four dif- 
ferent periods of the year, as near the equinoxes and the sol- 
stices, according to the following directions. The latitude sup- 
posed is 41°. 

Lksson I. — For the middle of September, from 8 to 10 o'clock. 
At 8 o'clock Scorpio is near setting in the S. W., Antares being 
10° high. The bow of Sagittarius is seen on the eastern mar- 
gin of the Milky Way, the arrow being directed to a point a 
little below Antares. At 9 o'clock, the horns of the Goat come 
upon the meridian ; and at 10 o'clock, the western shoulder of 
Aquarius. The other shoulder, and the figure Y in the arm, 
may also be easily found from the descriptionsgiven on p. 277 ; 
also, the Pentagon, in Pisces, and Fomalhaut (the Southern 
Pish), a solitary bright star far in the south, only 16° above 
the horizon. The head of Aries appears in the east, and the 
Pleiades are but little above the horizon, while Aldebaran is 
just rising. Returning now to the west (at 10 o'clock), the 
Crown is seen a little north of west, about 20° high ; Lyra is 
30° west of the zenith ; the Swan is nearly overhead : and fol- 
lowing down the Milky Way, the Eagle is seen on its eastern 
margin over against Lyra on the western ; and the Dolphin, a 
little eastward of the Eagle, and as far above the horns of Cap- 
ricornus, as the latter are above the southern horizon. Follow- 
ing on east of the meridian, the great square in Pegasus may 
next be identified ; and since the northeastern corner of the 
square is in the head of Andromeda, this constellation may 
next be learned ; and then Perseus and Auriga, which appear 
still further east. Directly north of Perseus, is Cassiopeia's 



286 FIXED STARS. 

chair; and next to that we may take the Pole-star, the Little 
Bear, and the Great Bear, the Dipper only "being traced for 
the present. Commencing now at the tail of the Dragon, we 
may trace round this figure between the two Bears to the head, 
which brings us back to Lyra and the head of Hercules. The 
boundaries of this constellation, and of Ophiuchus, which lies 
south of it, will end the first lesson. 

Lesson II. — For the middle of December, from 7 to 10 o'clock. 
Of the constellations of the Zodiac, Taurus and Gemini are 
now favorably situated for observation in the east. At 7 o'clock, 
the tail of Cetus just reaches the meridian, its head being seen 
below the feet of Aries. Orion is just risen in the S. E. At 
9 o'clock, just above the western horizon, are seen in succession 
from south to north, Aquarius, the Dolphin, the Eagle, the 
Lyre, and the Dragon's head. Between the Eagle and the 
Lyre, at a little higher altitude, we perceive the Swan, flying 
directly downward. Between the tail of the Swan and the 
Pole-star, is Cepheus ; and from the pole, along the meridian, 
we trace Cassiopeia, the feet of Andromeda, the head of Aries, 
and the neck of the Whale. At 10 o'clock, Perseus has reached 
the meridian, the star Algol, in the head of Medusa, being di- 
rectly overhead. The Pleiades are but little eastward of the 
zenith ; and following along south from the pole, at the inter- 
val of from one to two hours east of the meridian, we may 
trace in succession, Camelopard, Auriga, Taurus, Orion, and 
the Hare. Turning along the eastern horizon, we find Canis 
Major, Monoceros, Canis Minor, the head of Hydra (just rising), 
Cancer, Leo, the sickle just appearing about 3° north of the 
east point. Leo Minor and Ursa Major complete the survey; 
and we may now advantageously trace out the various parts 
of the Great Bear, as described on p. 278 ; the two stars com- 
posing its hindmost paw being scarcely above the horizon. 

Lesson III. — For the middle of March, from 8 to 10 o'clock. 
At 8 o'clock, we see the Twins nearly overhead, and Procyon 
and Sirius, at different intervals, toward the south. Along the 
west we recognize the neck and head of the whale, the head of 
Aries, and the head of Andromeda ; next above these, Orion, 
Taurus, Perseus, Cassiopeia, and Cepheus ; and north of the 



DOUBLE STARS. 287 

head of Orion, we see Auriga and Camelopard. In the S. "W., 
Hydra is now fully displayed ; and following on north, we ob- 
tain fine views of the Greater and the Lesser Lion, and the Great 
Bear. At 9 o'clock, Crater and Corvus appear in the S. E., on 
the back of Hydra ; Virgo extends from Leo down to the hori- 
zon, Spica Virginia being about 5° high ; and north of Virgo, 
we trace in succession Coma Berenices, Cor Caroli, Bootes, with 
Arct'urus, and the Crown lying far in the N. E. 

Lesson IV. — For the middle of June, from 9 to 10 o'clock. 
At 9 o'clock, Bootes, Corona Borealis, the head of Libra, the 
Serpent, and Scorpio, lie along on either side of the meridian. 
Castor and Pollux are just setting, and Leo is about an hour 
high. East of Leo, Virgo is seen extending along toward the 
meridian, Spica being about 30° above the southern horizon. 
North of Leo and Virgo we recognize Leo Minor, Coma Bere- 
nices, Cor Caroli, and Ursa Major. At 10 o'clock, we trace 
along the eastern side of the meridian, Draco, Hercules, and 
Ophiuchus ; and east of these, the Lyre, the Eagle, Antinous, 
Sagittarius, and Capricornus. North of the Eagle, and round 
to the east, we find Cepheus and Cassiopeia, Andromeda rising 
in the northeast, Pegasus in the east, and Aquarius in the 
southeast. Thus we may advantageously complete a review 
of the constellations. 



CHAPTEE II 



DOUBLE STARS TEMPORARY STARS — VARIABLE STARS CLUSTERS 

AND NEBULAE. 

418. The view hitherto taken of the starry heavens presents 
little that is new, since most of the Constellations, visible in 
our latitude, and the most conspicuous of the individual stars, 
have been known from antiquity. But the objects to be de- 
scribed in the present chapter, are chiefly such as have been 
discovered by modern astronomy, aided by the powerful tele- 
scopes which, since the time of Sir William Herschel, have been 
directed to the heavens. Different orders and systems of stars 



288 FIXED STARS. 

have been brought to light, and a new and still more wonder- 
ful class of bodies, called Nebulae, have been reached in the 
depths of the stellar universe. 

419. The introduction into practical astronomy of Her- 
scheTs great Forty-feet Reflector, in 1789, was a great event 
in the study of the stars. This instrument in its previous 
humble forms had been very little employed upon the stars, 
they being supposed to be too remote for its powers, which 
seemed only suited to nearer worlds, as the sun and planets. 
It was not, however, an increase of magnifying power that 
was wanted for researches on these distant objects, but 
an increase of light, by which a few scattered rays sent to 
us from bodies hidden in the depths of space, might be col- 
lected in such numbers, and directed into the eye, as would 
render visible objects otherwise invisible, not because they do 
not transmit us any light, but because not enough of what they 
transmit enters the small pupil of the eye for the purposes of 
distinct vision. Telescopes of great aperture, therefore, by col- 
lecting a large beam of light and conveying it to the eye, 
greatly enlarge the powers of this organ, and enable it to pen- 
etrate porportionally further into the most distant regions of 
the universe. Sir "W". Herschel himself made wonderful prog- 
ress in the knowledge of the starry heavens, and by his own 
researches discovered a large portion of those bodies which we 
are now to describe ; and his son, Sir John Herschel, has cul- 
tivated, with great success, the same field, and especially by a 
residence of five years at the Cape of Good Hope, devoted as- 
siduously to observations with large instruments, has greatly 
augmented our knowledge of the stellar systems of the south- 
ern hemisphere. Moreover, telescopes of still greater power 
than that of the elder Herschel, and especially instruments ca- 
pable of nicer angular measurements, have recently enriched' 
the department of practical astronomy. The most remarkable 
of these are the grand Reflector constructed by Lord Rosse, an 
Irish nobleman, and the great Refractors belonging respect- 
ively to the Pulkova and Cambridge Observatories. Lord 
Rosse's telescope considerably exceeds in dimensions and in 
power the forty-feet reflector of Sir "W. Herschel, being 50 feet 
in focal length, and having a diameter of 6 feet, whereas that 



DOUBLE STABS. 289 

of the Herschelian telescope was only 4 feet. This unexampled 
magnitude makes this instrument superior to all others in light, 
and fits it pre-eminently for observations on the most remote 
and obscure celestial objects, such as the faintest nebulae. But 
its unwieldy size, and its liability to loss of power, by the tar- 
nishing or temporary blurring of the great speculum, will 
render it far less available for actual research than the great 
refractors which come into competition with it. Until recently, 
it was thought impossible to form perfect achromatic object- 
glasses of more than about five inches diameter; but they 
have been successively enlarged, until we can no longer set 
bounds to the dimensions which they may finally assume. The 
Pulkova telescope (at St. Petersburg) has a clear aperture of 
about 15 inches, and a focal length of 22 feet. The telescope 
recently acquired by Harvard University, is perhaps the finest 
refractor hitherto constructed. It was made by the same 
artists, and upon the same scale with that, but its performances 
are thought even to exceed those of the Pulkova instrument. 
We now proceed to review some of the discoveries among the 
stars, which the researches made with such instruments as the 
foregoing have brought to light. 

DOUBLE STABS. 

420. Double Stabs are those which appear single to the 
naked eye, but are resolved into two by the telescope ; or if 
not visible to the naked eye, are seen in the telescope very close 
together. Sometimes three or more stars are found in this 
near connection, constituting triple or multiple stars.* Castor, 
for example, when seen by the naked eye, appears as a single 
star ; but in a telescope, even of moderate powers, it is resolved 
into two stars, between the third and fourth magnitudes, with- 
in 5 /; of each other. These two stars are of nearly equal size, 
but frequently one is exceedingly small in comparison with the 
other, resembling a satellite near its primary, although in dis- 
tance, in light, and in other characteristics, each has all the at- 
tributes of a star, and the combination, therefore, cannot be 
that of a planet with a satellite. The distance between these 

* See several figures of double and multiple stars, in Plate III. at the end of 
the volume, 

19 



290 FIXED STAKS. 

objects varies from a fraction of a second to thirty-two seconds. 
In some cases, the extreme closeness, and the exceeding 
minuteness of double stars, require, for their separation, the 
best telescope, united with the most acute powers of observa- 
tion. Indeed, certain of these objects are regarded as the 
severest tests both of the excellence of the instrument, and of 
the skill of the observer. 

421. When Sir "William Herschel began his observations 
on double stars, about the year 1780, he was acquainted with 
only 4. By his own researches he extended the number to 
2,400. Sir John Herschel, Sir James South, and M. Struve, 
the great Russian astronomer, prosecuted the same line of re- 
search ; and when Sir John Herschel left England for the 
Cape of Good Hope, in 1833, the whole number of double stars 
enrolled was 3,346 ; and this number was increased, by that 
eminent astronomer, by adding those of the southern hemi- 
sphere, to 5,542. It appears, therefore, that the number of 
double stars considerably exceeds all the stars visible to the 
naked eye. In some instances, this proximity arises undoubt- 
edly from the two members lying nearly in the same line of 
vision, and therefore being projected very near to each other on 
the face of the sky ; but in most cases the double stars are 
proved to have a physical relation to each other, and are there- 
fore said to be physically double, while the former are said to 
be optically double. There is no longer any doubt that among 
the stars are separate systems, in which two, three, and even 
in one instance at least, six stars are bound together in rela- 
tions of mutual dependence, suns with suns, as the members 
of the solar system compose an individual province in the 
great empire of nature. A star in Orion's sword (Theta 
Orionis) has been for some time known as a quadruple star, 
the members of which form a small trapezium ; and recent ob- 
servations have detected in two of these, severally, companions 
of extreme minuteness, the whole composing a figure like the 
following : 



TEMPORARY AND VARIABLE STARS. 291 

Many of the double stars are distinguished by the compo- 
nents exhibiting different colors, often finely contrasted with 
each other ; as orange with blue or green, yellow with blue, 
and white with purple. Gamma Andromedse is a close double 
star, the components of which are both green. Insulated stars 
of a red color, almost as deep as that of blood, occur in many 
parts of the heavens, but no green or blue star of any decided 
hue has ever been noticed unassociated with a companion 
brighter than itself.* 

422. TEMPORARY STARS. 

Temporary Stars are new stars which have suddenly made 
their appearance, and after a certain interval, as suddenly dis- 
appeared, and returned no more. It was the appearance of a 
new star of this kind, 125 years before the Christian era, that 
prompted Hipparchus to form a catalogue of the stars, the first 
on record. Such also was the star which suddenly shone out, 
a. d. 389, in the Eagle, as bright as Venus, and after remain- 
ing three weeks, disappeared entirely. At other periods, at 
distant intervals, similar phenomena have presented them- 
selves. Thus the appearance of a new star in 1572 was so sud- 
den, that Tycho Brahe, returning home one evening, was sur- 
prised to find a collection of country people gazing at a star 
which he was sure did not exist half an hour before. It was 
then as bright as Sirius, and continued to increase until it sur- 
passed Jupiter when brightest, and was visible at mid-day. In 
a month it began to diminish, and in three months afterward 
it had entirely disappeared. Some stars are now missing which 
were registered in the older catalogues. In one instance, at 
least (that of Neptune), the supposed star has proved to have 
been a planet. 

423. VARIABLE STARS. 

Variable Stars are those which undergo a periodical change 
of brightness. One of the most remarkable is the star Mira, in 
the neck of the Whale (Omicron Ceti). It appears once in 11 
months, remains at its greatest brightness about a fortnight, 
being then, on some occasions, equal to a star of the second 

* Herschel. 



292 FIXED STARS. 

magnitude. It then decreases about three months, until it be- 
comes completely invisible, and remains so about five months, 
when it again becomes visible, and continues increasing during 
the remaining three months of its period. 

Another very remarkable variable star is Algol (fl Persei). 
It is suddenly visible as a star of the second magnitude, and 
continues such for 2d. lih., when it begins rapidly to diminish 
in splendor, and in about BJ hours is reduced to the fourth 
magnitude. It then begins again to increase, and in 3£ hours 
more, is restored to its usual brightness, going through all its 
changes in less than three days. This remarkable law of vari- 
ation appears strongly to suggest the revolution round it of 
some opaque body, which, when interposed between us and 
Algol, cuts off a large portion of its light. It is (says Sir 
J. Herschel) an indication of a high degree of activity in regions 
where, but for such evidence, we might conclude all to be life- 
less. Our sun requires almost nine times this period to perform 
a revolution on its axis. On the other hand, the periodic time 
of an opaque revolving body, sufficiently large, which would 
produce a similar temporary obscuration of the sun, seen from 
a fixed star, would be less than fourteen hours. 

The duration of these periods is extremely various. While 
that of j3 Persei, above mentioned, is less than three days, 
others are more than a year, and others many years. 

424. CLUSTERS AND NEBULAE. 

In various parts of the firmament are seen large groups or 
Clusters, which, either by the naked eye, or by the aid of the 
smallest telescope, are perceived to consist of a great number 
of small stars. Such are the Pleiades, Coma Berenices, and 
Prsesepe, or the Beehive in Cancer. The Pleiades, or /Seven 
Stars, as they are called, in the neck of Taurus, is the most 
conspicuous cluster. When we look directly at this group, we 
can not distinguish more than six stars, but by turning the eyes 
a little to one side,* we discover that there are many more. 

* Indirect vision is far more delicate than direct. Thus we can see the Zodiacal 
Light or a Cornet's tail much more distinctly and of greater length, if, instead 
of looking directly at it, we turn the eyes to various points near it, the attention all 
the while being given to the object itself. 



CLUSTERS AND NEBULA. 293 

The telescope only can, however, display the real magnificence 
of the Pleiades. (See Plate III., Fig. 1.) Coma Berenices has 
fewer stars, but they are of a larger class than those which com- 
pose the Pleiades. The Beehive, or Nebula of Cancer, is one 
of the finest objects of this kind for a small telescope, being, by 
its aid, converted into a rich congeries of shining points. A 
cluster in the sword-handle of Perseus, below Cassiopeia's chair, 
though but a dim speck to the naked eje, is a very elegant 
object to a large telescope, being separated into bright and 
beautiful stars, embracing several distinct subordinate clusters 
of exceedingly minute stellar points. The head of Orion af- 
fords an example of another cluster, though less remarkable 
than the others. 

425. Nebula are faint misty objects seen in various parts 
of the firmament, always maintaining a fixed position, which 
resemble comets, or patches of fog. The Galaxy, or Milky 
Way, presents a constant succession of large nebulae. Of the 
individual nebulae, seen by the naked eye, the most conspicu- 
ous is that near the girdle of Andromeda. It is the oldest 
known nebula, having attracted the attention of star-gazers 
as early as the beginning of the tenth century,* although it is 
commonly said to have been discovered by Simon Marius, in 
1612. No powers of the telescope have been able to resolve 
this into separate stars, although the great Cambridge tele- 
scope reveals a vast number of stars, more than 1,500, of various 
degrees of brightness, scattered over its surface; but these ap- 
pear not to belong to the nebula itself, which has hitherto af- 
forded no evidence of resolution, f Its dimensions are aston- 
ishingly great, since it covers a space of a quarter of a degree 
in diameter ; and we must bear in mind that, at such a dis- 
tance as the fixed stars, a space of 15' implies an immense ex- 
tent. Its figure is oval, and elliptical nebulae constitute a com- 
mon variety among the figures which these bodies exhibit. 
(See Plate III., Fig. 2, for a representation of the great nebula 
of Andromeda.) Another very common figure are the globular- 
nebulae. A grand specimen of this variety may be easily 



* Smyth's Cycle, ii., p. 15. 

f Memoirs of the Amer. Acad., vol. iii. 



294 FIXED STARS. 

found in the constellation Hercules, between Zeta and Eta. 
©raw a line from Lyra to Gemma of the Crown, and 3° above 
the center of that line will be the place of this nebula. When 
viewed with a small telescope, it exhibits only a circular cloud 
(Plate III., Fig. 3, a), but to a more powerful instrument it re- 
veals its real glories in a form truly exciting to the beholder 
(Fig. 3, b). About 4000 nebulae have been detected and de- 
scribed, of which about 1,700 have recently been added by 
Sir John Herschel, from his Results of Observations at the 
Cape of Good Hope Among the latter are two remarkable 
spots, well known to navigators, situated near the south pole, 
called Magellanic clouds by sailors, but by astronomers, the 
Nubecula Major and the Nubecula Minor. They are found 
to consist of a wonderful collection of nebulae, the greater em- 
bracing 278 nebulae, and the lesser 37. Both together com- 
pose a most magnificent assemblage. In the sword of Orion 
is a celebrated nebula, long known, which, until recently, had 
resisted all attempts to resolve it into stars ; but the great Re- 
flector of Lord Rosse, and more recently the great Refractor 
of the Cambridge Observatory, have succeeded in a partial 
resolution, at least, of this grand object, and have authorized 
the anticipation that, with a small increase of telescopic power, 
the whole will be shown to consist of an immense collection of 
exceedingly minute stars. 

These great telescopes, by the superior light they afford, dis- 
play their peculiar powers in this department of astronomy, 
and those astronomers who, for the first time, have gazed at 
these sidereal pictures as seen in the "Leviathan" of Lord 
Rosse, have expressed, in glowing terms, their mingled delight 
and astonishment. The perfect forms, and strange but sym- 
metrical configurations, exhibited by these instruments, of 
nebulae that were before seen of irregular or fantastic shapes, 
afford grounds for believing that such irregularities are often 
if not always owing to the objects being but partly developed. 
Thus the Crab Nebula of Lord Rosse (Plate III., Fig. 4) had 
been long known as a faint, ill-defined nebula of an elliptical 
shape ; but the higher powers of that instrument exhibit the 
before-concealed appendages which are essential to the com- 
pleteness of the figure. The Whirlpool Nebula of Rosse 
(Plate III., Fig. 5), when seen in separate parts, exhibited no 



CLUSTERS AND NEBULA. 295 

signs of order or symmetry ; but when viewed with the great 
Reflector, it develops the wonderful structure of a perfect 
spiral. 

426. Nebulae were formerly divided into two classes, re- 
solvable and irresolvable, the former term implying that the 
body was shown by the telescope to consist of stars, and the 
latter implying that the body is not composed of stars, but of a 
shining, cloudy kind of matter diffused throughout the mass. 
Astronomers, at present, include all resolvable nebulae under 
the head of clusters, appropriating the term nebulae exclusively 
to such of these bodies as have never been resolved. The 
question whether this distinction is not merely relative to the 
powers of the telescope, and whether, on the increase of these 
powers, this class of bodies would not all be resolved into stars, 
is not easily determined, since the same increase of telescopic 
power which converts existing nebulae into clusters, brings to 
light a greater number of those which are irresolvable. 

These remote objects of the universe occasionally exhibit 
traces of that regard to beauty which everywhere, in these 
nether worlds, characterizes the works of the Creator. In the 
Cross, a brilliant constellation of the southern hemisphere, for 
example, is a cluster surrounding the star Kappa Crucis, which 
consists of about 110 stars from the seventh magnitude down- 
ward, eight of the more conspicuous of which are colored with 
various shades of red, green, and blue, so as to give to the 
whole the appearance of a rich piece of jewelry. 

427. Nebulous stars are such as exhibit a sharp and bril- 
liant star, surrounded by a disk or atmosphere of nebulous 
matter. These atmospheres in some cases, present a circular, 
in others an oval figure ; and in certain instances, the nebula 
consists of a long, narrow, spindle-shaped ray, tapering away 
at both ends to points. Annular Nebulae (Ring-shaped) are 
among the rarest objects in the heavens. The most conspicu- 
ous of this class is in the constellation Lyra, between the stars 
Beta and Gamma, about 6° S. E. of Alpha Lyrae. This re- 
markable object is believed to be in fact a resolvable nebula 
or cluster, and yet the greatest powers of the telescope hitherto 
applied have only effected such changes as are regarded as 



296 FIXED STAKS. 

giving signs of resolvability, but its perfect resolution has not 
"been attained. Should it be achieved by an increased power 
of the instrument, astronomers look for a splendid coronet of 
stars, more glorious f perhaps, than any thing hitherto discovered 
in the starry heavens. 

Planetary Wehulce constitute another variety, and are very 
remarkable objects. They have, as their name imports, exactly 
the appearance of planets. Whatever may be their nature, 
they must be of enormous magnitude. One of them is to be 
found in the parallel of v Aquarii, and about 5m. preceding 
that star. Its apparent diameter is about 20". Another in 
the constellation Andromeda, presents a visible disk of 12", 
perfectly defined and round. Granting these objects to be 
equally distant from us with the stars, their real dimensions 
must be such as, on the lowest computation, would fill the or- 
bit of Uranus. It is no less evident that, if they be solid 
bodies, of a solar nature, the intrinsic splendor of their surfaces 
must be almost infinitely inferior to that of the sun. A circular 
portion of the sun's disk, subtending an angle of 20", would give 
a light equal to 100 full moons ; while the objects in question 
are hardly, if at all, discernible with the naked eye.* 

428. The Milky Way, or Galaxy, is a well-known lumi- 
nous zone, encircling the sphere nearly in the direction of a 
great circle. Near the Swan, in the northern sky, it is seen 
to be divided into two bands, which remain asunder for 150°, 
and then reunite. The Galaxy owes its peculiar appearance 
to the blended light of myriads of small stars too minute to be 
individually recognized by the naked eye, but which are seen 
in their true character by a telescope of only moderate powers. 
Sir William Herschel estimated, that, on one occasion, in forty- 
one minutes, no less than 258,000 stars passed through the 
small field of his telescope.f In approaching the border of 
the Milky Way, there is found a regular but rapid increase in 
the number of stars, even before entering the limits of the lu- 
minous zone itself. Sir J. Herschel computes the whole num- 



* Herschel. 

f Plate II., Fig. 1, exhibits a telescopic view of a part of the southern portion 
of the Milky Way. 



MOTIONS OF THE FIXED STARS. 297 

ber of stars in the Milky Way 2Xfive and a half 'million s, in- 
cluding such only as are visible in his twenty-feet reflector. 
The Galaxy is supposed to be one of the numerous nebulse, 
and the sun of our own solar system to be one of the stars 
which compose it. It appears comparatively large to us, and 
extends entirely round the heavens, only because we are in 
the midst of it, and see it projected in different directions 
from us. 



CHAPTER III. 

MOTIONS OF THE FIXED STARS DISTANCES NATURE. 

429. In 1803, Sir William Herschel first determined and 
announced to the world, that there exist among the stars sepa- 
rate systems, composed of two stars, revolving about each 
other in regular orbits. These he denominated Binary Stars, 
to distinguish them from other double stars where no such 
motion is detected, and whose proximity to each other may 
possibly arise from casual juxtaposition, or from one being in 
the range of the other. At present, more than a hundred of 
the binary stars are known, and as the number of such revo- 
lutions known among the double stars is constantly increasing 
as the times of comparison increase, it may be anticipated that, 
in after ages, so large a proportion of all the double stars will 
be found to possess this character, as to authorize the belief 
that they universally consist of subordinate systems, of which 
the members have a revolution around a common center of 
gravity. The periodic times of the binary stars are very 
various. While some (as £ Herculis, and t\ Coronas) complete 
their revolutions in 30 or 40 years, others (as y Virginis) re- 
quire more than 170, and others still (as 65 Piscium) take up 
the long period of 3000 years.* Their orbits are in general 
more eccentric than those of the planets. That of Gamma 
Virginis, including the relative positions of the two components 

* Smyth's Cycle, i., p. 300. 



298 FIXED STARS. 

from 1837 to 1860, is figured on Plate IT. as drawn by Mr. E. 
P. Mason, in 1810.* 

430. The revolutions of the binary stars have assured us of 
this most interesting fact, that the law of gravitation extends to 
the fixed stars. Before these discoveries, we could not decide, 
except by a feeble analogy, that this law transcended the 
bounds of the solar system. Indeed, our belief of the fact 
rested more upon our idea of unity of design in all the works 
of the Creator, than upon any certain proof; but the revolu- 
tion of one star around another, in obedience to forces which 
must be similar to those that govern the solar system, estab- 
lishes the grand conclusion, that the law of gravitation is truly 
the law of the material universe. 

We have the same evidence (says Sir John Herschel) of the 
revolutions of the binary stars about each other, that we have of 
those of Saturn and Uranus about the sun ; and the correspond- 
ence between their calculated and observed places in such elon- 
gated ellipses, must be admitted to carry with it a proof of the 
prevalence of the Newtonian law of gravity in their systems, of 
the very same nature and cogency as that of the calculated and 
observed places of comets round the center of our own system. 

But (he adds) it is not with the revolutions of bodies of a 
planetary or cometary nature round a solar center that we are 
now concerned ; it is with that of sun around sun, each, per- 
haps, accompanied with its train of planets and their satellites, 
closely shrouded from our view by the splendor of their re- 
spective suns, and crowded into a space, bearing hardly a 
greater proportion to the enormous interval which separates 
them, than the distances of the satellites of our planets from 
their primaries, bear to their distances from the sun itself. 

431. Some of the fixed stars appear to have a Proper Mo- 
tion, or a real motion in space. 

e Sir John Herschel had computed the orbit of y Virginis, and had given it at 
625 years. Mason, from a discussion of all the observations, published to the 
date of 1838, combined with his own of 1840, found that this period was too 
great, and assigned as the true period 171 years, which is now acknowledged by 
the highest authorities, and even by Herschel himself, to be nearly its real time 
of revolution. 



MOTIONS OF THE FIXED IYTARS. 299 

The apparent change of place in the stars arising from the 
precession of the equinoxes, the nutation of the earth's axis, the 
diminution of the obliquity of the ecliptic, and the aberration 
of light, have been already mentioned ; but after all these cor- 
rections are made, changes of place still occur, which cannot 
result from any changes in the earth, but must arise from 
changes in the stars themselves. Such motions are called the 
proper motions of the stars. Nearly 2000 years ago, Hippar- 
chus and Ptolemy made the most accurate determinations in 
their power of the relative situations of the stars, and their ob- 
servations have been transmitted to us in Ptolemy's Almagest ; 
from which it appears that the stars retain at least very nearly 
the same places now as they did at that period. Still, the 
more accurate methods of modern astronomers, have brought 
to light minute changes in the places of certain stars which 
force upon us the conclusion, either that our solar system causes 
an apparent displacement of certain stars, by a motion of its 
own in space, or that they have themselves a proper motion. 
Possibly, indeed, both these causes may operate. 

432. If the sun, and of course the earth which accompanies 
him, is actually in motion, the fact may become manifest from 
the apparent approach of the stars in the region which he is 
leaving, and the recession of those which lie in the part of the 
heavens toward which he is traveling. Were two groves of 
trees situated on a plain at some distance apart, and we should 
go from one to the other, the trees before us would gradually 
appear further and further asunder, while those we left behind 
would appear to approach each other. Some years since, Sir 
William Herschel supposed he had detected changes of this 
kind among two sets of stars in opposite points of the heavens, 
and announced that the solar system was in motion toward a 
point in the constellation Hercules.* As, for many years after 
this announcement, other astronomers failed to find evidence 
of such a motion of the solar system, the doctrine was generally 
discredited, until, within a few years, new and very refined 
researches have been instituted by several of the most eminent 
astronomers, which have fully confirmed the observations ol 



* Phil. Trans., 1783, 1805, and 1806. 



300 FIXED STARS. 

Herschel. The great Russian astronomer, Struve, by a com- 
parison of the best observations, finds the exact point toward 
which the solar system is moving is in a line which joins the 
two stars * and /x Herculis,* — a point which can be easily found 
on the celestial globe, and thence transferred to the heavens. 
(Right ascension 259°, declination 34i°.) The researches of 
the younger Struve have conducted him to the velocity with 
which the solar system is moving in space. For having found 
that the arc traversed by the sun in a year is // .3392, if viewed 
at the mean distance of the stars of the first magnitude, and 
having previously ascertained that the mean parallax of the 
stars of this class amounts to 0".2d9j he infers that the space 
through which the sun moves annually is 154,000,000 miles. 
Great as this space is, yet it may be remarked that it is only 
about one-fourth that traversed by the earth in its revolution 
around the sun. Within the comparatively short period 
during which these observations on the solar motion have been 
continued, the direction appears rectilinear ; but all analogy 
leads to the belief that it is in fact a. motion of revolution, al- 
though on account of the immense size of the orbit, and, con- 
sequently, its small curvature, many years will be requisite in 
order to determine the deviation from the line of the tangent.f 

433. When we reflect on the immense distance of the 
stars, we may readily believe that they may be in fact in rapid 
motion, and yet appear quiescent ; as a distant ship, under full 
sail, appears at rest, although actually moving at the rate of 
ten knots an hour. Thus we have seen above that a motion of 
the sun in space, as seen from the nearest fixed stars, would 
make it describe an arc of only about one-third of a second 
annually, although traversing a space of 154 millions of miles. 
But a small change in the place of a star in a single year may, 
in a long series of years, accumulate to a very sensible amount. 
For example, the latitudes of the three bright stars, Sirius, 
Arcturus, and Aldebaran, were determined by Hipparchus 130 
years before the Christian era, and their assigned places are 
transmitted to us in the Almagest of Ptolemy. About the 
year 1700, Dr. Halley found that these stars had, during the 

* Etudes d'Astron. Stellaire, p. 108. f Grant's Hist. Phys. Ast., p. 557. 



DISTANCES OF THE FIXED STARS. 301 

interval of nearly 2000 years, moved southerly through the 
spaces respectively of 37', 42', and 33'. The immense pains 
that have of late years been bestowed upon catalogues of the 
stars, and especially of particular portions of the heavens, with 
the view of furnishing to after ages the most accurate data for 
comparison, will enable future astronomers to study the proper 
motions of the stars with far greater advantages than the pres- 
ent generation enjoys. In most cases where a proper motion 
in certain stars has been suspected, its annual amount has been 
so small, that many years are required to assure us that the 
effect is not owing to some other than a real progressive motion 
in the stars themselves ; but in a few instances the fact is too 
obvious to admit of any doubt. A small star in the leg of the 
Great Bear has an annual motion away from the neighboring 
stars of T' '; and the two stars 61 Cygni, which are nearly equal, 
have remained constantly at the same, or nearly at the same 
distance of 15" for at least fifty years past. Meanwhile they 
have shifted their local situation in the heavens M 23", the 
annual proper motion of each star being 5 ".3, by which quan- 
tity this system is every year carried along in some unknown 
path, by a motion which for many centuries must be regarded 
as uniform and rectilinear. A greater proportion of the double 
stars than of any other indicate proper motions, especially the 
binary stars, or those which have a revolution around each 
other. Among stars not double, and no way differing from 
the rest in any other obvious particular, ^ Cassiopeiae has a 
proper motion, amounting to nearly 4" annually ; and another 
obscure star has been recently found to have a motion of 
nearly S"> 

434. DISTANCES OF THE FIXED STARS. 

It has long been considered one of the highest problems that 
can be proposed to the human mind, to measure the distance 
to any of the fixed stars. Nothing more would be necessary 
than to measure the horizontal parallax, if it were possible ; 
but this is now known to be less than the 70,000th part of a 
second for the nearest star, and therefore utterly inappreciable 
by any method of measurement. For measuring the distances 

* Herschel's Outlines (Ed. 1851). 



302 FIXED STASS. 

of the sun and planets, the diameter of the earth furnishes the 
base line (Art. 87). The length of this line being known, and 
likewise the horizontal parallax of the body whose distance is 
sought, we readily obtain the distance by the solution of a 
right-angled triangle (Art. 80, Fig. 6). But any star viewed 
from the opposite sides of the earth would appear from both 
stations to occupy precisely the same situation in the celestial 
sphere, and of course it would exhibit no horizontal parallax. 
But astronomers have endeavored to find a parallax in some of 
the fixed stars, by taking the diameter of the earth's orbit as a 
base line. Yet even a change of position, amounting to 190 
millions of miles, has, until within a few years, proved insuffi- 
cient to alter the apparent place of a single fixed star, from 
which it was concluded that the fixed stars have not even any 
annual parallax ; or that the angle subtended by the semi-di- 
ameter of the earth's orbit, at the nearest fixed star, is insensible. 
The errors to which instrumental measurements are subject, 
arising from defects of the instruments themselves, from errors 
of refraction, of aberration, of precession, of nutation, and from 
imperfections of observation, are such, that the angular deter- 
minations of celestial arcs, it was supposed, could not be Fig 8 q 
relied on to less than \" ; and the change of place in any c 
star that had been examined for parallax being less than 
one second when viewed at opposite extremities of the 
earth's orbit, the conclusion was, that the parallax of the 
fixed stars, if any exist, is too minute ever to be measured 
by instruments. According to this, the diameter of the 
earth's orbit, when viewed from the nearest fixed star, 
would be insensible ; the spider-line of the telescope 
would more than cover it. 

Taking, however, the annual parallax at I", let ah 
(Fig. 80) represent the radius of the earth's orbit, and c 
a fixed star, the angle at c being 1", and the angle at b 
a right angle ; then, 

Sin 1" : Kad : : 1 : 200,000, nearly. 

Hence the hypotenuse of a triangle whose vertical ff ' — * 
angle is 1", is about 200,000 times the base; consequently, the 
distance in question must exceed 95,000,000 x 200,000 = 
190,000,000 x 100,000, or one hundred thousand times one 
hundred and ninety millions of miles. Of a distance so vast 



DISTANCES OF THE FIXED STARS. 303 

we can form no adequate conceptions, and attempt to measure 
it only by the time that light (which moves more than 192,000 
miles per second) would take to traverse it. Now, 

192,000 : Is. : : 19,000,000,000,000 : 3.1 years. 

435. After many fruitless and delusory efforts to measure 
the immense interval that separates us from the fixed stars, the 
great Prussian astronomer, Bessel, in the year 1838, determined 
this interesting and important element, by observations on a 
double star in the Swan (61 Cygni). This star was selected 
for the following reasons : first, it was known to have a great 
proper motion (Art. 433), indicating a comparatively great 
proximity to our system ; secondly, situated as it is among the 
circumpolar stars, observations could be made upon it nearly 
every night in the year ; and thirdly, the great number of small 
stars in the immediate neighborhood, furnished the opportu- 
nity of selecting favorable stationary points from which (inas- 
much as these more remote objects might be considered as en- 
tirely devoid of parallax) any changes of place in the nearer, 
in consequence of an annual parallax, might be readily esti- 
mated. By observations of the last degree of refinement, con- 
ducted for a period of several years, a parallax was decisively 
indicated, amounting to about one-third of a second ; or, more 
exactly, to 0".3483, implying a distance of 592,200 times the 
mean distance of the earth from the sun, or a space which it 
would take light, moving at the rate of twelve millions of miles 
per minute, nine and a quarter years to traverse. Perhaps the 
best way to conceive of this distance, is to compare it with the 
dimensions of the solar system, as represented by the diagram 
(note, p. 178). The distance from the sun to Neptune being 
30 feet, 61 Cygni should be placed 110 miles off. Thus isolated 
are the systems of the universe from each other. 

The observations of Bessel enabled him to estimate also the 
period of revolution of the two stars composing the binary sys- 
tem of 61 Cygni, and the dimensions of the orbit, and he found 
the periodic time about 540 years, and the length of the orbit 
about two and a half times that of Uranus. Knowing the dis- 
tance and proper motion of the star, we can now obtain its 
velocity, so far as it is perpendicular to our line of vision. This 
is found to be about forty-four miles per second — more than 



304 FIXED STARS. 

double that of the earth in its orbit — amounting to about one 
thousand millions of miles per annum. 

On account of the smallness of the supposed parallax thus 
found, it would not be unreasonable still to entertain a linger- 
ing suspicion, that it is nothing more than the unavoidable im- 
perfection of instrumental measurements, as proved to be the 
case in previous attempts to find the same element ; but the 
most satisfactory evidence which the world can have that such 
is not the fact in the present instance, but that the parallax is 
truly found, is that the most celebrated astronomers of the age, 
after rigorous scrutiny, have acknowledged the reality and 
soundness of the determination. Our confidence that the par- 
allax of 61 Cygni was truly determined by Bessel, is strength- 
ened by the fact that a separate determination recently made 
by Peters at the Pulkova Observatory, gives almost precisely 
the same result, that of Bessel being 0".348, and that of Peters 
0".349. In the case of several stars still more distant, the par- 
allax has been found, with more or less probability, but with 
sufficient to command the general confidence of astronomers. 
Thus, the parallax of Arcturus, Alpha Lyrse, and Polaris, were 
also found by Peters to be respectively 0".127, /7 .123, 0".067, 
that of the Pole-star being only one-fifth as great as that of 61 
Cygni ; and, consequently, if light would require 9^ years to 
come from that star, it would require more than 46 years to 
come to us from the Pole-star. A star in the southern hemi- 
sphere (a Centauri) indicates a parallax of about 1", and hence 
appears at present the nearest of the fixed stars. 

436. NATURE OF THE STARS. 

The stars are bodies greater than our earth. If this were not 
the case they could not be visible at such an immense distance. 
Dr. Wollaston, a distinguished English philosopher, attempted 
to estimate the magnitudes of certain of the fixed stars from 
the light which they afford. By means of an accurate photom- 
eter (an instrument for measuring the relative intensities of 
light) he compared the light of Sirius with that of the sun. 
He next inquired how far the sun must be removed from us 
in order to appear no brighter than Sirius. He found the dis- 
tance to be 141,400 times its present distance. But Sirius is 



NATURE OF THE STARS. 305 

more than 200,000 times as far off as the sun (Art. 434). 
Hence he inferred that, upon the lowest computation, Sirius 
must actually give out twice as much light as the sun ; or 
that, in point of splendor, Sirius must be at least equal to two 
suns. Indeed, he has rendered it probable that the light of 
Sirius is equal to fourteen suns. 

437. The fixed stars are suns. "We have already seen that 
they are large bodies ; that they are immensely further off 
than the furthest planet ; that they shine by their own light, 
as is evident by the nature of the light as tested by polarization ; 
in short, that their appearance is, in all respects, the same as the 
sun would exhibit if removed to the region of the stars. Hence 
we infer that they are bodies of the same kind with the sun. 

438. We are justified therefore by a sound analogy, in con- 
cluding that the stars were made for the same end as the sun, 
namely, as the centers of attraction to other planetary worlds, 
to which they severally dispense light and heat. Although 
the starry heavens present, in a clear night, a spectacle of in- 
effable grandeur and beauty, yet it must be admitted that the 
chief purpose of the stars could not have been to adorn the 
night, since by far the greater part of them are wholly invisi- 
ble to the naked eye ; nor as landmarks to the navigator, for 
only a very small proportion of them are adapted for this pur- 
pose; nor, finally, 'to influence the earth by their attractions, 
since their distance renders such an effect entirely insensible. 
If they are suns, and if they exert no important agencies upon 
our world, but are bodies evidently adapted to the same pur- 
pose as our sun, then it is as rational to suppose that they were 
made to give light and heat, as that the eye was made for see- 
ing and the ear for hearing. It is obvious to inquire next, to 
what they dispense these gifts, if not to planetary worlds ; and 
why to planetary worlds, if not for the use of percipient 
beings ? We are thus led, almost inevitably, to the idea of a 
Plurality of Worlds ; and the conclusion is forced upon us, 
that the spot which the Creator has assigned to us is but an 
humble province of his boundless empire.* 



* See this argument, in its full extent, in Dick's Celestial Scenery. 
20 



CHAPTEE IT. 

OF THE SYSTEM OF THE WORLD. 

439. The arrangement of all the bodies that compose the ma- 
terial universe, and their relations to each other, constitutes the 
System of the World. 

It is otherwise called the Mechanism of the Heavens ; and, 
indeed, in the System of the World, we figure to ourselves a 
machine, all the parts of which have a mutual dependence, and 
conspire to one great end. " The machines that are first in- 
vented (says Adam Smith) to perform any particular move- 
ment, are always the most complex ; and succeeding artists 
generally discover that with fewer wheels and with fewer prin- 
ciples of motion than had originally been employed, the same 
effects may be more easily produced. The first systems, in the 
same manner, are always the most complex ; and a particular 
connecting chain or principle is generally thought necessary to 
unite every two seemingly disjointed appearances ; but it often 
happens that one great connecting principle is afterward found 
to be sufficient to bind together all the discordant phenomena 
that occur in a whole species of things." This remark is 
strikingly applicable to the origin and progress of systems of 
astronomy. , 

440. From the visionary notions which are generally under- 
stood to have been entertained on this subject by the ancients, 
we are apt to imagine that they knew less than they actually 
did of the truths of astronomy. But Pythagoras, who lived 
500 years before the Christian era, was acquainted with many 
important facts in our science, and entertained many opinions 
respecting the System of the World which are now held to be 
true. Among other things well known to Pythagoras were 
the following : 

1. The principal constellations. These had begun to be 
formed in the earliest ages of the world. Several of them 
bearing the same names as at present are mentioned in the 
writings of Hesiod and Homer; and the "sweet influences of 



SYSTEM OF THE WORLD. 307 

the Pleiades" and the " bands of Orion," are beautifully alluded 
to in the Book of Job. 

2. Eclipses. Pythagoras knew both the causes of eclipses 
and how to predict them ;* not indeed in the accurate manner 
now employed, but by means of the Saros (Art. 233). 

3. Pythagoras had divined the true system of the world, 
holding that the sun, and not the earth (as was generally held 
by the ancients, even for many years after Pythagoras), is the 
center around which all the planets revolve, and that the stars 
are so many suns, each the center of a system like our own.f 
Among lesser things, he knew that the earth is round ; that its 
surface is naturally divided into five zones; and that the eclip- 
tic is inclined to the equator. He also held that the earth 
revolves daily on its axis, and yearly around the sun ; that the 
galaxy is an assemblage of small stars ; and that it is the same 
luminary, namely, Yenus, that constitutes both the morning 
and the evening star, whereas all the ancients before him had 
supposed that each was a separate planet, and accordingly the 
morning star was called Lucifer, and the evening star Hesper- 
us.;): He held also that the planets were inhabited, and even 
went so far as to calculate the size of some of the animals in 
the moon.§ Pythagoras was so great an enthusiast in music, 
that he not only assigned to it a conspicuous place in his system 
of education, but even supposed* the heavenly bodies them- 
selves to be arranged at distances corresponding to the diatonic 
scale, and imagined them to pursue their sublime march to 
notes created by their own harmonious movements, called the 
" music of the spheres ;" but he maintained that this celestial 
concert, though loud and grand, is not audible to the feeble 
organs of man, but only to the gods. 

441. With few exceptions, however, the opinions of Pythag- 
oras on the System of the World, were founded in truth. Yet 
they were rejected by Aristotle and by most succeeding astron- 
omers down to the time of Copernicus, and in their place was 
substituted the doctrine of Crystalline Spheres, first taught by 



° Long's Astronomy, ii., p. 671. 

f Library of Useful Knowledge, History of Astronomy. 

% Long's Ast., ii., p. 673. § Edin. Encyclopaedia 



308 SYSTEM OF THE WORLD. 

Eudoxus. According to this system, the heavenly bodies are 
set like gems in hollow solid orbs, composed of crystal so pel- 
lucid that no anterior orb obstructs in the least the view of any 
of the orbs that lie behind it. The sun and the planets have 
each its separate orb ; but the fixed stars are all set in the same 
grand orb ; and beyond this is another still, the Primwn Mo- 
hile, which revolves daily from east to west, and carries along 
with it all the other orbs. Above the whole spreads the 
Grand Einpyrean, or third heavens, the abode of perpetual 
serenity.* 

To account for the planetary motions, it was supposed that 
each of the planetary orbs, as well as that of the sun, has a 
motion of its own eastward, while it partakes of the common 
diurnal motion of the starry sphere. Aristotle taught that these 
motions are effected by a tutelary genius of each planet, resid- 
ing in it, and directing its motions, as the mind of man directs 
his motions. 

442. On coming down to the time of Hipparchus, who 
flourished about 150 years before the Christian era, we meet 
with astronomers who acquired far more accurate knowledge 
of the celestial motions. Hipparchus was in possession of in- 
struments for measuring angles, and knew how to resolve 
spherical triangles. He ascertained the length of the year 
within 6m. of the truth. He discovered the eccentricity of the 
solar orbit (although he supposed the sun actually to move 
uniformly in a circle, but the earth to be placed out of the 
center), and the positions of the sun's apogee and perigee. He 
formed very accurate estimates of the obliquity of the ecliptic 
and of the precession of the equinoxes. He computed the exact 
period of the synodic revolution of the moon, and the inclina- 
tion of the lunar orbit ; discovered the motion of her node and 
of her line of apsides ; and made the first attempts to ascertain 
the horizontal parallaxes of the sun and moon. 

Such was the state of astronomical knowledge when Ptolemy 
wrote the Almagest, in which he has transmitted to us an en- 
cyclopaedia of the astronomy of the ancients. 

* Long's Ast., ii., p. 640 ; Robinson's Mech. Phil., ii., p. 83 ; Gregory's Ast., 
p. 132; Playfair's Dissertations, p. 118. 



THE PTOLEMAIC SYSTEM. 309 

443. The systems of the world which have been most cele- 
brated are three — the Ptolemaic, the Tychonic, and the Coper- 
niean. We shall conclude this part of our work with a con- 
cise statement and discussion of each of these systems of the 
Mechanism of the Heavens. 



THE PTOLEMAIC SYSTEM. 

444. The doctrines of the Ptolemaic System were not origi- 
nated by Ptolemy ; but being digested by him out of materials 
furnished by various hands, it has come down to us under the 
sanction of his name. 

According to this system, the earth is the center of the uni- 
verse, and all the heavenly bodies daily revolve around it from 
east to west. In order to explain the planetary motions, Ptol- 
emy had recourse to deferents and epicycles, — an explanation 
devised by Apollonius, one of the greatest geometers of an- 
tiquity.* He conceived that, in the circumference of a circle, 
having the earth for its center, there moves the center of an- 
other circle, in the circumference of which the planet actually 
revolves. The circle surrounding the earth was called the 
deferent, while the smaller circle, whose center was always in 
the periphery of the deferent, was called the epicycle. The mo- 
tion in each was supposed to be uniform. Lastly, it was con- 
ceived that the motion of the center of the epicycle in the cir- 
cumference of the deferent, and of the deferent itself, are in 
opposite directions, the first being toward the east, and the 
second toward the west. 

445. But these views will be better understood from a dia- 
gram. Therefore, let ABC (Fig. 81) represent the deferent, E 
being the earth a little out of the center. Let abc represent 
the epicycle, having its center at v, on the periphery of the def- 
erent. Conceive the circumference of the deferent to be carried 
about the earth every twenty-four hours in the order of the 
letters ; and at the same time, let the center v of the epicycle 
abed, have a slow motion in the opposite direction, and let a 
body revolve in this circle in the direction abed. Then it will 

* Play fair, Dissertation Second, p. 119. 



310 



SYSTEM OF THE WORLD. 



be seen that the body would actually describe the looped 
curves klmnop ; that it would appear stationary at I and m, 
and at n and o ; that its motion would be direct from k to I, 
and then retrograde from I to m ; direct again from m to n y 

Fig. 81. 




and retrograde from n to o. Thus, suppose Mercury to be sit- 
uated at h in its epicycle. By the revolution of the deferent, 
it would be carried along with the other heavenly bodies 
around the earth from left to right, every twenty -four hours ; 
but, meanwhile, the center of the epicycle shifting its place 
slowly from right to left, while Mercury was moving from h to 
Cj c itself would change its place to r, and therefore the path of 
the planet would be in the cycloid al arc hr. Again, while 
Mercury was passing through cda, the point c would be still 
moving eastward, which would have the effect apparently to 
compress the lower half of the epicycle into the looped curve 
nor ; and as on this side the motion in the epicycle is in the 
same direction with that of the deferent, but at a slower rate, 
the apparent path is much shorter than where, as on the other 
side, the two motions conspire" 



THE TYCHONIC SYSTEM. 311 

446. Sueli a deferent and epicycle may be devised for each 
planet as will fully explain all its ordinary motions ; but it is 
inconsistent with the phases of Mercury and Yenus, which 
being between us and the sun on both sides of the epicycle, 
would present their dark sides toward ns in both these posi- 
tions, whereas at one of the conjunctions they are seen to shine 
with full face.* It is moreover absurd to speak of a geometri- 
cal center, which has no bodily existence, moving around the 
earth on the circumference of another circle ; and hence some 
suppose that the ancients merely assumed this hypothesis as 
affording a convenient geometrical representation of the phe- 
nomena — a diagram simply, without conceiving the system to 
have any real existence in nature. 

447. The objections to the Ptolemaic system, in general, are 
the following : First, it is a mere hypothesis, having no evi- 
dence in its favor, except that it explains the phenomena. This 
evidence is insufficient of itself, since it frequently happens that 
each of two hypotheses, directly opposite to each other, will 
explain all the known phenomena. But the Ptolemaic system 
does not even do this, as it is inconsistent with the phases of 
Mercury and Yenus, as already observed. Secondly, now that 
we are acquainted with the distances of the remoter planets, 
and especially of the fixed stars, the swiftness of motion implied 
in a daily revolution of the starry firmament around the earth, 
renders such a motion wholly incredible. Thirdly, the centrif- 
ugal force that would be generated in these bodies, especially 
in the sun, renders it impossible that they can continue to re- 
volve around the earth as a center. 

These reasons are sufficient to show the absurdities of the 
Ptolemaic System of the World. 

THE TYCHONIC SYSTEM. 

448. Tycho Brahe, like Ptolemy, placed the earth in the 
center of the universe, and accounted for the diurnal motions 
in the same manner as Ptolemy had done, namely, by an actual 
revolution of the whole host of heaven around the earth every 
twenty-four hours. But he rejected the scheme of deferents 

* Vince's Complete System, i. r p. 96. 



312 SYSTEM OF THE WOULD. 

and epicycles, and held that the moon revolves about the earth 
as the center of her motions ; that the sun, and not the earth, 
is the center of the planetary motions ; and that the sun, ac- 
companied by the planets, moves around the earth once a year, 
somewhat in the manner that we now conceive of Jupiter and 
his satellites as revolving around the sun. The system of 
Tycho serves to explain all the common phenomena of the 
planetary motions, but it is encumbered with the same objec- 
tions as those that have been mentioned as resting against the 
Ptolemaic system, namely, that it is a mere hypothesis ; that 
it implies an incredible swiftness in the diurnal motions ; and 
that it is inconsistent with the known laws of universal gravi- 
tation. But if the heavens do not revolve, the earth must, and 
this brings us to the system of Copernicus. 

THE COPEKNTCAN SYSTEM. 

449. Copernicus was born at Thorn, in Prussia, in 1473. 
The system that bears his name was the fruit of forty years of 
intense study and meditation upon the celestial motions. As 
already mentioned (Art. 6), it maintains (1), That the apparent 
diurnal motion of the heavenly bodies, from east to west, is 
owing to the real revolution of the earth on its own axis from 
west to east ; and (2), That the sun is the center around which 
the earth and planets all revolve from west to easl It rests 
on the following arguments : 

First, the earth revolves on its own axis. 

1. Because this supposition is vastly more simple. 

2. It is agreeable to analogy, since all the other planets that 
afford any means of determining the question, are seen to re- 
volve on their axes. 

3. The spheroidal figure of the earth is the figure of equilib- 
rium, that results from a revolution on its axis. 

4. The diminished weight of bodies at the equator, indicates 
a centrifugal force arising from such a revolution. 

5. Bodies let fall from a high eminence, fall eastward of their 
oase, indicating that higher objects have greater velocity of 
rotation than lower ones. 

6. The precession of the equinoxes is explained by the earth's 
rotation on its axis. 



THE COPERNICAN SYSTEM. CI Z 

Secondly, the planets, including the earth, revolve about the sun . 

1. The phases of Mercury and Yenus are precisely such as 
would result from their circulating around the sun in orbits 
within that of the earth ; but they are never seen in opposition, 
as they would be if they circulated around the earth. 

2. The superior planets do indeed revolve around the earth ; 
but they also revolve around the sun, as is evident from their 
phases and from the known dimensions of their orbits ; and 
that the sun, and not the earth, is the center of their motions, 
is inferred from the greater symmetry of their motions as re- 
ferred to the sun than as referred to the earth, and especially 
from the laws of gravitation, which forbid our supposing that 
bodies so much larger than the earth, as some of these bodies 
are, can circulate permanently around the earth, the latter re- 
maining all the while at rest. 

3. The annual motion of the earth itself is indicated also by 
the most conclusive arguments. For, first, since all the planets 
with their satellites, and the comets, revolve about the sun, 
analogy leads us to infer the same respecting the earth and its 
satellite. Secondly, the motions of the satellites, as those of 
Jupiter and Saturn, indicate that it is a law of the solar system 
that the smaller bodies revolve about the larger. Thirdly, the 
direction of the periodical meteors of November, which, in a 
majority of cases, is from east to west, indicates the motion of 
the earth from west to east. Lastly, the aberration of light 
affords a sensible proof of the motion of the earth, since that 
phenomenon indicates both a progressive motion of light, and 
a motion of the earth from west to east. (Art. 195.) 

45 O It only remains to inquire whether there subsist high- 
er orders of relations between the stars themselves. The as- 
semblage of bodies in clusters, as in the Pleiades, and still 
more, as in the great nebula of Hercules (Art. 425), implies 
mutual relations constituting for each a system within itself; 
and the analogies of all that portion of the heavenly bodies, 
whose motions fall within our observation, and the known uni- 
formity of the laws of nature, conspire to prove that those re- 
lations are maintained by revolutions around a common center. 
What theory would lead us to expect, we actually see exempli- 
fied in the revolutions of the binary stars (Art. 430), and in 



314: SYSTEM OF THE WOULD. 

the motion of the sun himself with his attendant worlds (Art. 
432). The Nebulm also compose peculiar systems, in which 
the members seem associated in mutual relations, and separated 
from all the other heavenly bodies, each composing an " island 
universe." Thus we ascend from the lower to the higher com- 
binations, according to a uniform, plan, so characteristic of as- 
cending orders in every department of nature. Beginning 
with the relation between the earth and its satellite, we see it 
sustained by the prevalence of forces which subject it to Kepler's 
laws and the law of universal gravitation. We see the same 
principles carried out on a larger scale, but exactly on the 
samej??««, in the system of Jupiter and his satellites, and in 
the respective systems of Saturn, Uranus, and Neptune. From 
this lowest order of combination, composed of planets and 
their satellites, we ascend to the next higher order, consisting 
of suns and planets, in which the same plan is exemplified on 
a still grander scale, but without any change in its peculiar 
features. We next ascend still higher to the third order, as in 
the binary stars, where sun revolves around sun, upon the same 
unvarying plan as before seen in these nearer worlds. At 
present, observation leads us to no higher point of the scale in 
the structure of the universe ; but the mind of man, obtaining 
from these lower systems a knowledge of the plan on which 
the universe is built, goes forward to complete the grand ma- 
chine. A bold attempt has recently been made by Maedler, 
an eminent European astronomer, to fix the center, around 
which not only our sun, but all the stars of our firmament re- 
volve. It must evidently be such a point, that the known 
proper motions detected among the fixed stars will conform to 
it, like the motions of the planets around the sun. He places 
that center in the Pleiades, or, more exactly, in Alcyone, the 
central star of the Pleiades, which body is therefore denomi- 
nated the Central Sun.* The proofs of this remarkable hy- 
pothesis are deemed, too incomplete at present to command en- 
tire assent ; but the method of investigation pursued by this 
distinguished astronomer, opens a new field of observation and 
of speculation, and promises to lend a new interest to inquiries 
into the mechanism of the universe. 



* Plate III., 1. 



STRUCTURE OF THE UNIVERSE. 315 

451. This fact being now established, that the stars are 
immense bodies like the sun, and that they are subject to the 
laws of gravitation, we can not conceive how they can be pre- 
served from falling into final disorder and ruin, unless they 
move in harmonious concert like the members of the solar sys- 
tem. Otherwise, those that are situated on the confines of 
creation, being retained by no forces from without, while they 
are subject to the attraction of all the bodies within, must leave 
their stations, and move inward with accelerated velocity, and 
thus all the bodies in the universe would at length fall together 
in the common center of gravity. The immense distances at 
which the stars are placed from each other would indeed delay 
such a catastrophe ; but such must be the ultimate tendency of 
the material world, unless sustained in one harmonious system 
by nicely-adjusted motions.* To leave entirely out of view our 
confidence in the wisdom and preserving goodness of the 
Creator, and reasoning merely from what we know of the sta- 
bility of the solar system, we should be justified in inferring 
that other worlds are not subject to forces which operate only 
to hasten their decay, and to involve them in final ruin. 

We conclude, therefore, that the material universe is one great 
system ; that the combination of planets with their satellites 
constitutes the first or lowest order of worlds ; that next to 
these, planets are linked to suns ; that these are bound to other 
suns, composing a still higher order in the scale of being; and, 
finally, that all the different systems of worlds move around 
their common center of gravity. 

* Robison's Physical Astronomy. 



PLATE II. 
NEBULA AND DOUBLE STARS. 




1. Castor. 2. y Leonis. 3. 39 Drac. 4. X Opli. 5. 1 1 Monoo. (>. £Cancri. 




Revolutions of y Virginia. 



71 



PLATE IN. 
CLUSTERS AND NEBUL 




